What is the general solution for 3tanh(2θ)=5sech(θ)+1 ?

The general solution for 3tanh(2θ)=5sech(θ)+1 is θ=ln(4.82043…)

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3tanh(2θ)=5sech(θ)+1

Solution

3 tanh (2 θ) = 5 sech (θ) + 1

Solution

θ = ln (4.82043 \dots)

Degrees

θ = 90.11846 \dots^{\circ}

Solution steps

3 tanh (2 θ) = 5 sech (θ) + 1

Rewrite using trig identities

3 tanh (2 θ) = 5 sech (θ) + 1

Use the Hyperbolic identity:

tanh (x) = \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}

3 \cdot \frac{e ^{2 θ} - e ^{- 2 θ}}{e ^{2 θ} + e ^{- 2 θ}} = 5 sech (θ) + 1

Use the Hyperbolic identity:

sech (x) = \frac{2}{e ^{x} + e ^{- x}}

3 \cdot \frac{e ^{2 θ} - e ^{- 2 θ}}{e ^{2 θ} + e ^{- 2 θ}} = 5 \cdot \frac{2}{e ^{θ} + e ^{- θ}} + 1

3 \cdot \frac{e ^{2 θ} - e ^{- 2 θ}}{e ^{2 θ} + e ^{- 2 θ}} = 5 \cdot \frac{2}{e ^{θ} + e ^{- θ}} + 1

3 \cdot \frac{e ^{2 θ} - e ^{- 2 θ}}{e ^{2 θ} + e ^{- 2 θ}} = 5 \cdot \frac{2}{e ^{θ} + e ^{- θ}} + 1 : θ = ln (4.82043 \dots)

3 \cdot \frac{e ^{2 θ} - e ^{- 2 θ}}{e ^{2 θ} + e ^{- 2 θ}} = 5 \cdot \frac{2}{e ^{θ} + e ^{- θ}} + 1

Apply exponent rules

3 \cdot \frac{e ^{2 θ} - e ^{- 2 θ}}{e ^{2 θ} + e ^{- 2 θ}} = 5 \cdot \frac{2}{e ^{θ} + e ^{- θ}} + 1

Apply exponent rule:

a^{b c} = (a^{b})^{c}

e^{2 θ} = (e^{θ})^{2}, e^{- 2 θ} = (e^{θ})^{- 2}, e^{- θ} = (e^{θ})^{- 1}

3 \cdot \frac{( e ^{θ} ) ^{2} - ( e ^{θ} ) ^{- 2}}{( e ^{θ} ) ^{2} + ( e ^{θ} ) ^{- 2}} = 5 \cdot \frac{2}{e ^{θ} + ( e ^{θ} ) ^{- 1}} + 1

3 \cdot \frac{( e ^{θ} ) ^{2} - ( e ^{θ} ) ^{- 2}}{( e ^{θ} ) ^{2} + ( e ^{θ} ) ^{- 2}} = 5 \cdot \frac{2}{e ^{θ} + ( e ^{θ} ) ^{- 1}} + 1

Rewrite the equation with

e^{θ} = u

3 \cdot \frac{( u ) ^{2} - ( u ) ^{- 2}}{( u ) ^{2} + ( u ) ^{- 2}} = 5 \cdot \frac{2}{u + ( u ) ^{- 1}} + 1

Solve

3 \cdot \frac{u ^{2} - u ^{- 2}}{u ^{2} + u ^{- 2}} = 5 \cdot \frac{2}{u + u ^{- 1}} + 1 : u \approx - 0.45284 \dots, u \approx 4.82043 \dots

3 \cdot \frac{u ^{2} - u ^{- 2}}{u ^{2} + u ^{- 2}} = 5 \cdot \frac{2}{u + u ^{- 1}} + 1

Refine

\frac{3 ( u ^{4} - 1 )}{u ^{4} + 1} = \frac{10 u}{u ^{2} + 1} + 1

Multiply by LCM

\frac{3 ( u ^{4} - 1 )}{u ^{4} + 1} = \frac{10 u}{u ^{2} + 1} + 1

Find Least Common Multiplier of

u^{4} + 1, u^{2} + 1 : (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

u^{4} + 1, u^{2} + 1

Lowest Common Multiplier (LCM)

Factor the expressions

Factor

u^{4} + 1 : (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

u^{4} + 1

u^{4} + 1 = (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

= (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Compute an expression comprised of factors that appear either in

(u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

u^{2} + 1

= (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Multiply by LCM=

(u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

\frac{3 ( u ^{4} - 1 )}{u ^{4} + 1} (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) = \frac{10 u}{u ^{2} + 1} (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) + 1 \cdot (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Simplify

\frac{3 ( u ^{4} - 1 )}{u ^{4} + 1} (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) = \frac{10 u}{u ^{2} + 1} (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) + 1 \cdot (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Simplify

\frac{3 ( u ^{4} - 1 )}{u ^{4} + 1} (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) : 3 (u + 1) (u - 1) (u^{2} + 1)^{2}

\frac{3 ( u ^{4} - 1 )}{u ^{4} + 1} (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 ( u ^{4} - 1 ) ( u ^{2} + 1 ) ( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 )}{u ^{4} + 1}

Factor

3 (u^{4} - 1) (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) : 3 (u + 1) (u - 1) (u^{2} + 1)^{2} (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

3 (u^{4} - 1) (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Factor

u^{4} - 1 : (u^{2} + 1) (u + 1) (u - 1)

u^{4} - 1

Rewrite

u^{4} - 1

(u^{2})^{2} - 1^{2}

u^{4} - 1

Rewrite

1

1^{2}

= u^{4} - 1^{2}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

u^{4} = (u^{2})^{2}

= (u^{2})^{2} - 1^{2}

= (u^{2})^{2} - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(u^{2})^{2} - 1^{2} = (u^{2} + 1) (u^{2} - 1)

= (u^{2} + 1) (u^{2} - 1)

Factor

u^{2} - 1 : (u + 1) (u - 1)

u^{2} - 1

Rewrite

1

1^{2}

= u^{2} - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

u^{2} - 1^{2} = (u + 1) (u - 1)

= (u + 1) (u - 1)

= (u^{2} + 1) (u + 1) (u - 1)

= 3 (u + 1) (u - 1) (u^{2} + 1)^{2} (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

= \frac{3 ( u + 1 ) ( u - 1 ) ( u ^{2} + 1 ) ^{2} ( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 )}{u ^{4} + 1}

u^{4} + 1 = (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

= \frac{3 ( u + 1 ) ( u - 1 ) ( u ^{2} + 1 ) ^{2} ( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 )}{( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 )}

Cancel

\frac{3 ( u + 1 ) ( u - 1 ) ( u ^{2} + 1 ) ^{2} ( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 )}{( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 )} : 3 (u + 1) (u - 1) (u^{2} + 1)^{2}

\frac{3 ( u + 1 ) ( u - 1 ) ( u ^{2} + 1 ) ^{2} ( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 )}{( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 )}

Cancel the common factor:

u^{2} + 2 u + 1

= \frac{3 ( u + 1 ) ( u - 1 ) ( u ^{2} + 1 ) ^{2} ( u ^{2} - 2 u + 1 )}{u ^{2} - 2 u + 1}

Cancel the common factor:

u^{2} - 2 u + 1

= 3 (u + 1) (u - 1) (u^{2} + 1)^{2}

= 3 (u + 1) (u - 1) (u^{2} + 1)^{2}

Simplify

\frac{10 u}{u ^{2} + 1} (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) : 10 u (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

\frac{10 u}{u ^{2} + 1} (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{10 u ( u ^{2} + 1 ) ( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 )}{u ^{2} + 1}

Cancel the common factor:

u^{2} + 1

= 10 u (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Simplify

1 \cdot (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) : (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

1 \cdot (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Multiply:

1 \cdot (u^{2} + 1) = (u^{2} + 1)

= (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

3 (u + 1) (u - 1) (u^{2} + 1)^{2} = 10 u (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) + (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

3 (u + 1) (u - 1) (u^{2} + 1)^{2} = 10 u (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) + (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

3 (u + 1) (u - 1) (u^{2} + 1)^{2} = 10 u (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) + (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Solve

3 (u + 1) (u - 1) (u^{2} + 1)^{2} = 10 u (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) + (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) : u \approx - 0.45284 \dots, u \approx 4.82043 \dots

3 (u + 1) (u - 1) (u^{2} + 1)^{2} = 10 u (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) + (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Expand

3 (u + 1) (u - 1) (u^{2} + 1)^{2} : 3 u^{6} + 3 u^{4} - 3 u^{2} - 3

3 (u + 1) (u - 1) (u^{2} + 1)^{2}

(u^{2} + 1)^{2} = u^{4} + 2 u^{2} + 1

(u^{2} + 1)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = u^{2}, b = 1

= (u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2}

Simplify

(u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2} : u^{4} + 2 u^{2} + 1

(u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (u^{2})^{2} + 2 \cdot 1 \cdot u^{2} + 1

(u^{2})^{2} = u^{4}

(u^{2})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= u^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= u^{4}

2 u^{2} \cdot 1 = 2 u^{2}

2 u^{2} \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 u^{2}

= u^{4} + 2 u^{2} + 1

= u^{4} + 2 u^{2} + 1

= 3 (u + 1) (u - 1) (u^{4} + 2 u^{2} + 1)

Expand

(u + 1) (u - 1) : u^{2} - 1

(u + 1) (u - 1)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = u, b = 1

= u^{2} - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= u^{2} - 1

= 3 (u^{2} - 1) (u^{4} + 2 u^{2} + 1)

Expand

(u^{2} - 1) (u^{4} + 2 u^{2} + 1) : u^{6} + u^{4} - u^{2} - 1

(u^{2} - 1) (u^{4} + 2 u^{2} + 1)

Distribute parentheses

= u^{2} u^{4} + u^{2} \cdot 2 u^{2} + u^{2} \cdot 1 + (- 1) u^{4} + (- 1) \cdot 2 u^{2} + (- 1) \cdot 1

Apply minus-plus rules

+ (- a) = - a

= u^{4} u^{2} + 2 u^{2} u^{2} + 1 \cdot u^{2} - 1 \cdot u^{4} - 1 \cdot 2 u^{2} - 1 \cdot 1

Simplify

u^{4} u^{2} + 2 u^{2} u^{2} + 1 \cdot u^{2} - 1 \cdot u^{4} - 1 \cdot 2 u^{2} - 1 \cdot 1 : u^{6} + u^{4} - u^{2} - 1

u^{4} u^{2} + 2 u^{2} u^{2} + 1 \cdot u^{2} - 1 \cdot u^{4} - 1 \cdot 2 u^{2} - 1 \cdot 1

u^{4} u^{2} = u^{6}

u^{4} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{4} u^{2} = u^{4 + 2}

= u^{4 + 2}

Add the numbers:

4 + 2 = 6

= u^{6}

2 u^{2} u^{2} = 2 u^{4}

2 u^{2} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u^{2} = u^{2 + 2}

= 2 u^{2 + 2}

Add the numbers:

2 + 2 = 4

= 2 u^{4}

1 \cdot u^{2} = u^{2}

1 \cdot u^{2}

Multiply:

1 \cdot u^{2} = u^{2}

= u^{2}

1 \cdot u^{4} = u^{4}

1 \cdot u^{4}

Multiply:

1 \cdot u^{4} = u^{4}

= u^{4}

1 \cdot 2 u^{2} = 2 u^{2}

1 \cdot 2 u^{2}

Multiply the numbers:

1 \cdot 2 = 2

= 2 u^{2}

1 \cdot 1 = 1

1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 1

= u^{6} + 2 u^{4} + u^{2} - u^{4} - 2 u^{2} - 1

Group like terms

= u^{6} + 2 u^{4} - u^{4} + u^{2} - 2 u^{2} - 1

Add similar elements:

u^{2} - 2 u^{2} = - u^{2}

= u^{6} + 2 u^{4} - u^{4} - u^{2} - 1

Add similar elements:

2 u^{4} - u^{4} = u^{4}

= u^{6} + u^{4} - u^{2} - 1

= u^{6} + u^{4} - u^{2} - 1

= 3 (u^{6} + u^{4} - u^{2} - 1)

Expand

3 (u^{6} + u^{4} - u^{2} - 1) : 3 u^{6} + 3 u^{4} - 3 u^{2} - 3

3 (u^{6} + u^{4} - u^{2} - 1)

Distribute parentheses

= 3 u^{6} + 3 u^{4} + 3 (- u^{2}) + 3 (- 1)

Apply minus-plus rules

+ (- a) = - a

= 3 u^{6} + 3 u^{4} - 3 u^{2} - 3 \cdot 1

Multiply the numbers:

3 \cdot 1 = 3

= 3 u^{6} + 3 u^{4} - 3 u^{2} - 3

= 3 u^{6} + 3 u^{4} - 3 u^{2} - 3

Expand

10 u (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) + (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) : 10 u^{5} + 10 u + u^{6} + u^{4} + u^{2} + 1

10 u (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) + (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Expand

10 u (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) : 10 u^{5} + 10 u

Expand

(u^{2} + 2 u + 1) (u^{2} - 2 u + 1) : u^{4} + 1

(u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Distribute parentheses

= u^{2} u^{2} + u^{2} (- 2 u) + u^{2} \cdot 1 + 2 u u^{2} + 2 u (- 2 u) + 2 u \cdot 1 + 1 \cdot u^{2} + 1 \cdot (- 2 u) + 1 \cdot 1

Apply minus-plus rules

+ (- a) = - a

= u^{2} u^{2} - 2 u^{2} u + 1 \cdot u^{2} + 2 u^{2} u - 22 uu + 1 \cdot 2 u + 1 \cdot u^{2} - 1 \cdot 2 u + 1 \cdot 1

Simplify

u^{2} u^{2} - 2 u^{2} u + 1 \cdot u^{2} + 2 u^{2} u - 22 uu + 1 \cdot 2 u + 1 \cdot u^{2} - 1 \cdot 2 u + 1 \cdot 1 : u^{4} + 1

u^{2} u^{2} - 2 u^{2} u + 1 \cdot u^{2} + 2 u^{2} u - 22 uu + 1 \cdot 2 u + 1 \cdot u^{2} - 1 \cdot 2 u + 1 \cdot 1

Group like terms

= u^{2} u^{2} - 2 u^{2} u + 1 \cdot u^{2} + 2 u^{2} u + 1 \cdot u^{2} - 22 uu + 1 \cdot 2 u - 1 \cdot 2 u + 1 \cdot 1

Add similar elements:

1 \cdot 2 u - 1 \cdot 2 u = 0

= u^{2} u^{2} - 2 u^{2} u + 1 \cdot u^{2} + 2 u^{2} u + 1 \cdot u^{2} - 22 uu + 1 \cdot 1

Add similar elements:

- 2 u^{2} u + 2 u^{2} u = 0

= u^{2} u^{2} + 1 \cdot u^{2} + 1 \cdot u^{2} - 22 uu + 1 \cdot 1

Add similar elements:

1 \cdot u^{2} + 1 \cdot u^{2} = 2 u^{2}

= u^{2} u^{2} + 2 u^{2} - 22 uu + 1 \cdot 1

u^{2} u^{2} = u^{4}

u^{2} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u^{2} = u^{2 + 2}

= u^{2 + 2}

Add the numbers:

2 + 2 = 4

= u^{4}

22 uu = 2 u^{2}

22 uu

Apply radical rule:

a a = a

22 = 2

= 2 uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= 2 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 u^{2}

1 \cdot 1 = 1

1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 1

= u^{4} + 2 u^{2} - 2 u^{2} + 1

Add similar elements:

2 u^{2} - 2 u^{2} = 0

= u^{4} + 1

= u^{4} + 1

= 10 u (u^{4} + 1)

Expand

10 u (u^{4} + 1) : 10 u^{5} + 10 u

10 u (u^{4} + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = 10 u, b = u^{4}, c = 1

= 10 u u^{4} + 10 u \cdot 1

= 10 u^{4} u + 10 \cdot 1 \cdot u

Simplify

10 u^{4} u + 10 \cdot 1 \cdot u : 10 u^{5} + 10 u

10 u^{4} u + 10 \cdot 1 \cdot u

10 u^{4} u = 10 u^{5}

10 u^{4} u

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{4} u = u^{4 + 1}

= 10 u^{4 + 1}

Add the numbers:

4 + 1 = 5

= 10 u^{5}

10 \cdot 1 \cdot u = 10 u

10 \cdot 1 \cdot u

Multiply the numbers:

10 \cdot 1 = 10

= 10 u

= 10 u^{5} + 10 u

= 10 u^{5} + 10 u

= 10 u^{5} + 10 u

= 10 u^{5} + 10 u + (u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Expand

(u^{2} + 1) (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) : u^{6} + u^{4} + u^{2} + 1

Expand

(u^{2} + 1) (u^{2} + 2 u + 1) : u^{4} + 2 u^{3} + 2 u^{2} + 2 u + 1

(u^{2} + 1) (u^{2} + 2 u + 1)

Distribute parentheses

= u^{2} u^{2} + u^{2} 2 u + u^{2} \cdot 1 + 1 \cdot u^{2} + 1 \cdot 2 u + 1 \cdot 1

= u^{2} u^{2} + 2 u^{2} u + 1 \cdot u^{2} + 1 \cdot u^{2} + 1 \cdot 2 u + 1 \cdot 1

Simplify

u^{2} u^{2} + 2 u^{2} u + 1 \cdot u^{2} + 1 \cdot u^{2} + 1 \cdot 2 u + 1 \cdot 1 : u^{4} + 2 u^{3} + 2 u^{2} + 2 u + 1

u^{2} u^{2} + 2 u^{2} u + 1 \cdot u^{2} + 1 \cdot u^{2} + 1 \cdot 2 u + 1 \cdot 1

Add similar elements:

1 \cdot u^{2} + 1 \cdot u^{2} = 2 u^{2}

= u^{2} u^{2} + 2 u^{2} u + 2 u^{2} + 1 \cdot 2 u + 1 \cdot 1

u^{2} u^{2} = u^{4}

u^{2} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u^{2} = u^{2 + 2}

= u^{2 + 2}

Add the numbers:

2 + 2 = 4

= u^{4}

2 u^{2} u = 2 u^{3}

2 u^{2} u

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u = u^{2 + 1}

= 2 u^{2 + 1}

Add the numbers:

2 + 1 = 3

= 2 u^{3}

1 \cdot 2 u = 2 u

1 \cdot 2 u

Multiply:

1 \cdot 2 = 2

= 2 u

1 \cdot 1 = 1

1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 1

= u^{4} + 2 u^{3} + 2 u^{2} + 2 u + 1

= u^{4} + 2 u^{3} + 2 u^{2} + 2 u + 1

= (u^{4} + 2 u^{3} + 2 u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Expand

(u^{4} + 2 u^{3} + 2 u^{2} + 2 u + 1) (u^{2} - 2 u + 1) : u^{6} + u^{4} + u^{2} + 1

(u^{4} + 2 u^{3} + 2 u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

Distribute parentheses

= u^{4} u^{2} + u^{4} (- 2 u) + u^{4} \cdot 1 + 2 u^{3} u^{2} + 2 u^{3} (- 2 u) + 2 u^{3} \cdot 1 + 2 u^{2} u^{2} + 2 u^{2} (- 2 u) + 2 u^{2} \cdot 1 + 2 u u^{2} + 2 u (- 2 u) + 2 u \cdot 1 + 1 \cdot u^{2} + 1 \cdot (- 2 u) + 1 \cdot 1

Apply minus-plus rules

+ (- a) = - a

= u^{4} u^{2} - 2 u^{4} u + 1 \cdot u^{4} + 2 u^{3} u^{2} - 22 u^{3} u + 1 \cdot 2 u^{3} + 2 u^{2} u^{2} - 22 u^{2} u + 2 \cdot 1 \cdot u^{2} + 2 u^{2} u - 22 uu + 1 \cdot 2 u + 1 \cdot u^{2} - 1 \cdot 2 u + 1 \cdot 1

Simplify

u^{4} u^{2} - 2 u^{4} u + 1 \cdot u^{4} + 2 u^{3} u^{2} - 22 u^{3} u + 1 \cdot 2 u^{3} + 2 u^{2} u^{2} - 22 u^{2} u + 2 \cdot 1 \cdot u^{2} + 2 u^{2} u - 22 uu + 1 \cdot 2 u + 1 \cdot u^{2} - 1 \cdot 2 u + 1 \cdot 1 : u^{6} + u^{4} + u^{2} + 1

u^{4} u^{2} - 2 u^{4} u + 1 \cdot u^{4} + 2 u^{3} u^{2} - 22 u^{3} u + 1 \cdot 2 u^{3} + 2 u^{2} u^{2} - 22 u^{2} u + 2 \cdot 1 \cdot u^{2} + 2 u^{2} u - 22 uu + 1 \cdot 2 u + 1 \cdot u^{2} - 1 \cdot 2 u + 1 \cdot 1

Group like terms

= u^{4} u^{2} - 2 u^{4} u + 1 \cdot u^{4} + 2 u^{3} u^{2} - 22 u^{3} u + 1 \cdot 2 u^{3} + 2 u^{2} u^{2} - 22 u^{2} u + 2 \cdot 1 \cdot u^{2} + 2 u^{2} u + 1 \cdot u^{2} - 22 uu + 1 \cdot 2 u - 1 \cdot 2 u + 1 \cdot 1

Add similar elements:

1 \cdot 2 u - 1 \cdot 2 u = 0

= u^{4} u^{2} - 2 u^{4} u + 1 \cdot u^{4} + 2 u^{3} u^{2} - 22 u^{3} u + 1 \cdot 2 u^{3} + 2 u^{2} u^{2} - 22 u^{2} u + 2 \cdot 1 \cdot u^{2} + 2 u^{2} u + 1 \cdot u^{2} - 22 uu + 1 \cdot 1

Add similar elements:

- 22 u^{2} u + 2 u^{2} u = - 2 u^{2} u

= u^{4} u^{2} - 2 u^{4} u + 1 \cdot u^{4} + 2 u^{3} u^{2} - 22 u^{3} u + 1 \cdot 2 u^{3} + 2 u^{2} u^{2} - 2 u^{2} u + 2 \cdot 1 \cdot u^{2} + 1 \cdot u^{2} - 22 uu + 1 \cdot 1

u^{4} u^{2} = u^{6}

u^{4} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{4} u^{2} = u^{4 + 2}

= u^{4 + 2}

Add the numbers:

4 + 2 = 6

= u^{6}

2 u^{4} u = 2 u^{5}

2 u^{4} u

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{4} u = u^{4 + 1}

= 2 u^{4 + 1}

Add the numbers:

4 + 1 = 5

= 2 u^{5}

1 \cdot u^{4} = u^{4}

1 \cdot u^{4}

Multiply:

1 \cdot u^{4} = u^{4}

= u^{4}

2 u^{3} u^{2} = 2 u^{5}

2 u^{3} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{3} u^{2} = u^{3 + 2}

= 2 u^{3 + 2}

Add the numbers:

3 + 2 = 5

= 2 u^{5}

22 u^{3} u = 2 u^{4}

22 u^{3} u

Apply radical rule:

a a = a

22 = 2

= 2 u^{3} u

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{3} u = u^{3 + 1}

= 2 u^{3 + 1}

Add the numbers:

3 + 1 = 4

= 2 u^{4}

1 \cdot 2 u^{3} = 2 u^{3}

1 \cdot 2 u^{3}

Multiply:

1 \cdot 2 = 2

= 2 u^{3}

2 u^{2} u^{2} = 2 u^{4}

2 u^{2} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u^{2} = u^{2 + 2}

= 2 u^{2 + 2}

Add the numbers:

2 + 2 = 4

= 2 u^{4}

2 u^{2} u = 2 u^{3}

2 u^{2} u

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u = u^{2 + 1}

= 2 u^{2 + 1}

Add the numbers:

2 + 1 = 3

= 2 u^{3}

2 \cdot 1 \cdot u^{2} = 2 u^{2}

2 \cdot 1 \cdot u^{2}

Multiply the numbers:

2 \cdot 1 = 2

= 2 u^{2}

1 \cdot u^{2} = u^{2}

1 \cdot u^{2}

Multiply:

1 \cdot u^{2} = u^{2}

= u^{2}

22 uu = 2 u^{2}

22 uu

Apply radical rule:

a a = a

22 = 2

= 2 uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= 2 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 u^{2}

1 \cdot 1 = 1

1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 1

= u^{6} - 2 u^{5} + u^{4} + 2 u^{5} - 2 u^{4} + 2 u^{3} + 2 u^{4} - 2 u^{3} + 2 u^{2} + u^{2} - 2 u^{2} + 1

Group like terms

= u^{6} - 2 u^{5} + 2 u^{5} + u^{4} - 2 u^{4} + 2 u^{4} + 2 u^{3} - 2 u^{3} + 2 u^{2} + u^{2} - 2 u^{2} + 1

Add similar elements:

2 u^{3} - 2 u^{3} = 0

= u^{6} - 2 u^{5} + 2 u^{5} + u^{4} - 2 u^{4} + 2 u^{4} + 2 u^{2} + u^{2} - 2 u^{2} + 1

Add similar elements:

- 2 u^{5} + 2 u^{5} = 0

= u^{6} + u^{4} - 2 u^{4} + 2 u^{4} + 2 u^{2} + u^{2} - 2 u^{2} + 1

Add similar elements:

2 u^{2} + u^{2} - 2 u^{2} = u^{2}

= u^{6} + u^{4} - 2 u^{4} + 2 u^{4} + u^{2} + 1

Add similar elements:

u^{4} - 2 u^{4} + 2 u^{4} = u^{4}

= u^{6} + u^{4} + u^{2} + 1

= u^{6} + u^{4} + u^{2} + 1

= u^{6} + u^{4} + u^{2} + 1

= 10 u^{5} + 10 u + u^{6} + u^{4} + u^{2} + 1

3 u^{6} + 3 u^{4} - 3 u^{2} - 3 = 10 u^{5} + 10 u + u^{6} + u^{4} + u^{2} + 1

Switch sides

10 u^{5} + 10 u + u^{6} + u^{4} + u^{2} + 1 = 3 u^{6} + 3 u^{4} - 3 u^{2} - 3

Subtract

3 u^{6} + 3 u^{4} - 3 u^{2} - 3

from both sides

10 u^{5} + 10 u + u^{6} + u^{4} + u^{2} + 1 - (3 u^{6} + 3 u^{4} - 3 u^{2} - 3) = 3 u^{6} + 3 u^{4} - 3 u^{2} - 3 - (3 u^{6} + 3 u^{4} - 3 u^{2} - 3)

Simplify

10 u^{5} + 10 u + u^{6} + u^{4} + u^{2} + 1 - (3 u^{6} + 3 u^{4} - 3 u^{2} - 3) = 3 u^{6} + 3 u^{4} - 3 u^{2} - 3 - (3 u^{6} + 3 u^{4} - 3 u^{2} - 3)

Simplify

10 u^{5} + 10 u + u^{6} + u^{4} + u^{2} + 1 - (3 u^{6} + 3 u^{4} - 3 u^{2} - 3) : - 2 u^{6} + 10 u^{5} - 2 u^{4} + 4 u^{2} + 10 u + 4

10 u^{5} + 10 u + u^{6} + u^{4} + u^{2} + 1 - (3 u^{6} + 3 u^{4} - 3 u^{2} - 3)

- (3 u^{6} + 3 u^{4} - 3 u^{2} - 3) : - 3 u^{6} - 3 u^{4} + 3 u^{2} + 3

- (3 u^{6} + 3 u^{4} - 3 u^{2} - 3)

Distribute parentheses

= - (3 u^{6}) - (3 u^{4}) - (- 3 u^{2}) - (- 3)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 3 u^{6} - 3 u^{4} + 3 u^{2} + 3

= 10 u^{5} + 10 u + u^{6} + u^{4} + u^{2} + 1 - 3 u^{6} - 3 u^{4} + 3 u^{2} + 3

Simplify

10 u^{5} + 10 u + u^{6} + u^{4} + u^{2} + 1 - 3 u^{6} - 3 u^{4} + 3 u^{2} + 3 : - 2 u^{6} + 10 u^{5} - 2 u^{4} + 4 u^{2} + 10 u + 4

10 u^{5} + 10 u + u^{6} + u^{4} + u^{2} + 1 - 3 u^{6} - 3 u^{4} + 3 u^{2} + 3

Group like terms

= u^{6} - 3 u^{6} + 10 u^{5} + u^{4} - 3 u^{4} + u^{2} + 3 u^{2} + 10 u + 1 + 3

Add similar elements:

u^{2} + 3 u^{2} = 4 u^{2}

= u^{6} - 3 u^{6} + 10 u^{5} + u^{4} - 3 u^{4} + 4 u^{2} + 10 u + 1 + 3

Add similar elements:

u^{4} - 3 u^{4} = - 2 u^{4}

= u^{6} - 3 u^{6} + 10 u^{5} - 2 u^{4} + 4 u^{2} + 10 u + 1 + 3

Add similar elements:

u^{6} - 3 u^{6} = - 2 u^{6}

= - 2 u^{6} + 10 u^{5} - 2 u^{4} + 4 u^{2} + 10 u + 1 + 3

Add the numbers:

1 + 3 = 4

= - 2 u^{6} + 10 u^{5} - 2 u^{4} + 4 u^{2} + 10 u + 4

= - 2 u^{6} + 10 u^{5} - 2 u^{4} + 4 u^{2} + 10 u + 4

Simplify

3 u^{6} + 3 u^{4} - 3 u^{2} - 3 - (3 u^{6} + 3 u^{4} - 3 u^{2} - 3) : 0

3 u^{6} + 3 u^{4} - 3 u^{2} - 3 - (3 u^{6} + 3 u^{4} - 3 u^{2} - 3)

Add similar elements:

3 u^{6} + 3 u^{4} - 3 u^{2} - 3 - (3 u^{6} + 3 u^{4} - 3 u^{2} - 3) = 0

= 0

- 2 u^{6} + 10 u^{5} - 2 u^{4} + 4 u^{2} + 10 u + 4 = 0

- 2 u^{6} + 10 u^{5} - 2 u^{4} + 4 u^{2} + 10 u + 4 = 0

Find one solution for

- 2 u^{6} + 10 u^{5} - 2 u^{4} + 4 u^{2} + 10 u + 4 = 0

using Newton-Raphson:

u \approx - 0.45284 \dots

- 2 u^{6} + 10 u^{5} - 2 u^{4} + 4 u^{2} + 10 u + 4 = 0

Newton-Raphson Approximation Definition

f (u) = - 2 u^{6} + 10 u^{5} - 2 u^{4} + 4 u^{2} + 10 u + 4

Find

f^{^{'}} (u) : - 12 u^{5} + 50 u^{4} - 8 u^{3} + 8 u + 10

\frac{d}{d u} (- 2 u^{6} + 10 u^{5} - 2 u^{4} + 4 u^{2} + 10 u + 4)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d u} (2 u^{6}) + \frac{d}{d u} (10 u^{5}) - \frac{d}{d u} (2 u^{4}) + \frac{d}{d u} (4 u^{2}) + \frac{d}{d u} (10 u) + \frac{d}{d u} (4)

\frac{d}{d u} (2 u^{6}) = 12 u^{5}

\frac{d}{d u} (2 u^{6})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{6})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 6 u^{6 - 1}

Simplify

= 12 u^{5}

\frac{d}{d u} (10 u^{5}) = 50 u^{4}

\frac{d}{d u} (10 u^{5})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 10 \frac{d}{d u} (u^{5})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 10 \cdot 5 u^{5 - 1}

Simplify

= 50 u^{4}

\frac{d}{d u} (2 u^{4}) = 8 u^{3}

\frac{d}{d u} (2 u^{4})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{4})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 4 u^{4 - 1}

Simplify

= 8 u^{3}

\frac{d}{d u} (4 u^{2}) = 8 u

\frac{d}{d u} (4 u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 4 \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 4 \cdot 2 u^{2 - 1}

Simplify

= 8 u

\frac{d}{d u} (10 u) = 10

\frac{d}{d u} (10 u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 10 \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 10 \cdot 1

Simplify

= 10

\frac{d}{d u} (4) = 0

\frac{d}{d u} (4)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= - 12 u^{5} + 50 u^{4} - 8 u^{3} + 8 u + 10 + 0

Simplify

= - 12 u^{5} + 50 u^{4} - 8 u^{3} + 8 u + 10

Let

u_{0} = 0

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 0.4 : Δ u_{1} = 0.4

f (u_{0}) = - 2 \cdot 0^{6} + 10 \cdot 0^{5} - 2 \cdot 0^{4} + 4 \cdot 0^{2} + 10 \cdot 0 + 4 = 4

f^{^{'}} (u_{0}) = - 12 \cdot 0^{5} + 50 \cdot 0^{4} - 8 \cdot 0^{3} + 8 \cdot 0 + 10 = 10

u_{1} = - 0.4

Δ u_{1} = ∣ - 0.4 - 0 ∣ = 0.4

Δ u_{1} = 0.4

u_{2} = - 0.45487 \dots : Δ u_{2} = 0.05487 \dots

f (u_{1}) = - 2 (- 0.4)^{6} + 10 (- 0.4)^{5} - 2 (- 0.4)^{4} + 4 (- 0.4)^{2} + 10 (- 0.4) + 4 = 0.478208

f^{^{'}} (u_{1}) = - 12 (- 0.4)^{5} + 50 (- 0.4)^{4} - 8 (- 0.4)^{3} + 8 (- 0.4) + 10 = 8.71488

u_{2} = - 0.45487 \dots

Δ u_{2} = ∣ - 0.45487 \dots - (- 0.4) ∣ = 0.05487 \dots

Δ u_{2} = 0.05487 \dots

u_{3} = - 0.45285 \dots : Δ u_{3} = 0.00201 \dots

f (u_{2}) = - 2 (- 0.45487 \dots)^{6} + 10 (- 0.45487 \dots)^{5} - 2 (- 0.45487 \dots)^{4} + 4 (- 0.45487 \dots)^{2} + 10 (- 0.45487 \dots) + 4 = - 0.01916 \dots

f^{^{'}} (u_{2}) = - 12 (- 0.45487 \dots)^{5} + 50 (- 0.45487 \dots)^{4} - 8 (- 0.45487 \dots)^{3} + 8 (- 0.45487 \dots) + 10 = 9.48821 \dots

u_{3} = - 0.45285 \dots

Δ u_{3} = ∣ - 0.45285 \dots - (- 0.45487 \dots) ∣ = 0.00201 \dots

Δ u_{3} = 0.00201 \dots

u_{4} = - 0.45284 \dots : Δ u_{4} = 3.93806 E - 6

f (u_{3}) = - 2 (- 0.45285 \dots)^{6} + 10 (- 0.45285 \dots)^{5} - 2 (- 0.45285 \dots)^{4} + 4 (- 0.45285 \dots)^{2} + 10 (- 0.45285 \dots) + 4 = - 0.00003 \dots

f^{^{'}} (u_{3}) = - 12 (- 0.45285 \dots)^{5} + 50 (- 0.45285 \dots)^{4} - 8 (- 0.45285 \dots)^{3} + 8 (- 0.45285 \dots) + 10 = 9.45147 \dots

u_{4} = - 0.45284 \dots

Δ u_{4} = ∣ - 0.45284 \dots - (- 0.45285 \dots) ∣ = 3.93806 E - 6

Δ u_{4} = 3.93806 E - 6

u_{5} = - 0.45284 \dots : Δ u_{5} = 1.47831 E - 11

f (u_{4}) = - 2 (- 0.45284 \dots)^{6} + 10 (- 0.45284 \dots)^{5} - 2 (- 0.45284 \dots)^{4} + 4 (- 0.45284 \dots)^{2} + 10 (- 0.45284 \dots) + 4 = - 1.39721 E - 10

f^{^{'}} (u_{4}) = - 12 (- 0.45284 \dots)^{5} + 50 (- 0.45284 \dots)^{4} - 8 (- 0.45284 \dots)^{3} + 8 (- 0.45284 \dots) + 10 = 9.45140 \dots

u_{5} = - 0.45284 \dots

Δ u_{5} = ∣ - 0.45284 \dots - (- 0.45284 \dots) ∣ = 1.47831 E - 11

Δ u_{5} = 1.47831 E - 11

u \approx - 0.45284 \dots

Apply long division:

\frac{- 2 u ^{6} + 10 u ^{5} - 2 u ^{4} + 4 u ^{2} + 10 u + 4}{u + 0.45284 \dots} = - 2 u^{5} + 10.90569 \dots u^{4} - 6.93863 \dots u^{3} + 3.14215 \dots u^{2} + 2.57708 \dots u + 8.83297 \dots

- 2 u^{5} + 10.90569 \dots u^{4} - 6.93863 \dots u^{3} + 3.14215 \dots u^{2} + 2.57708 \dots u + 8.83297 \dots \approx 0

Find one solution for

- 2 u^{5} + 10.90569 \dots u^{4} - 6.93863 \dots u^{3} + 3.14215 \dots u^{2} + 2.57708 \dots u + 8.83297 \dots = 0

using Newton-Raphson:

u \approx 4.82043 \dots

- 2 u^{5} + 10.90569 \dots u^{4} - 6.93863 \dots u^{3} + 3.14215 \dots u^{2} + 2.57708 \dots u + 8.83297 \dots = 0

Newton-Raphson Approximation Definition

f (u) = - 2 u^{5} + 10.90569 \dots u^{4} - 6.93863 \dots u^{3} + 3.14215 \dots u^{2} + 2.57708 \dots u + 8.83297 \dots

Find

f^{^{'}} (u) : - 10 u^{4} + 43.62278 \dots u^{3} - 20.81589 \dots u^{2} + 6.28430 \dots u + 2.57708 \dots

\frac{d}{d u} (- 2 u^{5} + 10.90569 \dots u^{4} - 6.93863 \dots u^{3} + 3.14215 \dots u^{2} + 2.57708 \dots u + 8.83297 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d u} (2 u^{5}) + \frac{d}{d u} (10.90569 \dots u^{4}) - \frac{d}{d u} (6.93863 \dots u^{3}) + \frac{d}{d u} (3.14215 \dots u^{2}) + \frac{d}{d u} (2.57708 \dots u) + \frac{d}{d u} (8.83297 \dots)

\frac{d}{d u} (2 u^{5}) = 10 u^{4}

\frac{d}{d u} (2 u^{5})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{5})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 5 u^{5 - 1}

Simplify

= 10 u^{4}

\frac{d}{d u} (10.90569 \dots u^{4}) = 43.62278 \dots u^{3}

\frac{d}{d u} (10.90569 \dots u^{4})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 10.90569 \dots \frac{d}{d u} (u^{4})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 10.90569 \dots \cdot 4 u^{4 - 1}

Simplify

= 43.62278 \dots u^{3}

\frac{d}{d u} (6.93863 \dots u^{3}) = 20.81589 \dots u^{2}

\frac{d}{d u} (6.93863 \dots u^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 6.93863 \dots \frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 6.93863 \dots \cdot 3 u^{3 - 1}

Simplify

= 20.81589 \dots u^{2}

\frac{d}{d u} (3.14215 \dots u^{2}) = 6.28430 \dots u

\frac{d}{d u} (3.14215 \dots u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 3.14215 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 3.14215 \dots \cdot 2 u^{2 - 1}

Simplify

= 6.28430 \dots u

\frac{d}{d u} (2.57708 \dots u) = 2.57708 \dots

\frac{d}{d u} (2.57708 \dots u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2.57708 \dots \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 2.57708 \dots \cdot 1

Simplify

= 2.57708 \dots

\frac{d}{d u} (8.83297 \dots) = 0

\frac{d}{d u} (8.83297 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= - 10 u^{4} + 43.62278 \dots u^{3} - 20.81589 \dots u^{2} + 6.28430 \dots u + 2.57708 \dots + 0

Simplify

= - 10 u^{4} + 43.62278 \dots u^{3} - 20.81589 \dots u^{2} + 6.28430 \dots u + 2.57708 \dots

Let

u_{0} = - 2

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 1.48484 \dots : Δ u_{1} = 0.51515 \dots

f (u_{0}) = - 2 (- 2)^{5} + 10.90569 \dots (- 2)^{4} - 6.93863 \dots (- 2)^{3} + 3.14215 \dots (- 2)^{2} + 2.57708 \dots (- 2) + 8.83297 \dots = 310.24761 \dots

f^{^{'}} (u_{0}) = - 10 (- 2)^{4} + 43.62278 \dots (- 2)^{3} - 20.81589 \dots (- 2)^{2} + 6.28430 \dots (- 2) + 2.57708 \dots = - 602.23740 \dots

u_{1} = - 1.48484 \dots

Δ u_{1} = ∣ - 1.48484 \dots - (- 2) ∣ = 0.51515 \dots

Δ u_{1} = 0.51515 \dots

u_{2} = - 1.06652 \dots : Δ u_{2} = 0.41831 \dots

f (u_{1}) = - 2 (- 1.48484 \dots)^{5} + 10.90569 \dots (- 1.48484 \dots)^{4} - 6.93863 \dots (- 1.48484 \dots)^{3} + 3.14215 \dots (- 1.48484 \dots)^{2} + 2.57708 \dots (- 1.48484 \dots) + 8.83297 \dots = 102.09660 \dots

f^{^{'}} (u_{1}) = - 10 (- 1.48484 \dots)^{4} + 43.62278 \dots (- 1.48484 \dots)^{3} - 20.81589 \dots (- 1.48484 \dots)^{2} + 6.28430 \dots (- 1.48484 \dots) + 2.57708 \dots = - 244.06592 \dots

u_{2} = - 1.06652 \dots

Δ u_{2} = ∣ - 1.06652 \dots - (- 1.48484 \dots) ∣ = 0.41831 \dots

Δ u_{2} = 0.41831 \dots

u_{3} = - 0.69341 \dots : Δ u_{3} = 0.37311 \dots

f (u_{2}) = - 2 (- 1.06652 \dots)^{5} + 10.90569 \dots (- 1.06652 \dots)^{4} - 6.93863 \dots (- 1.06652 \dots)^{3} + 3.14215 \dots (- 1.06652 \dots)^{2} + 2.57708 \dots (- 1.06652 \dots) + 8.83297 \dots = 34.94642 \dots

f^{^{'}} (u_{2}) = - 10 (- 1.06652 \dots)^{4} + 43.62278 \dots (- 1.06652 \dots)^{3} - 20.81589 \dots (- 1.06652 \dots)^{2} + 6.28430 \dots (- 1.06652 \dots) + 2.57708 \dots = - 93.66242 \dots

u_{3} = - 0.69341 \dots

Δ u_{3} = ∣ - 0.69341 \dots - (- 1.06652 \dots) ∣ = 0.37311 \dots

Δ u_{3} = 0.37311 \dots

u_{4} = - 0.21473 \dots : Δ u_{4} = 0.47868 \dots

f (u_{3}) = - 2 (- 0.69341 \dots)^{5} + 10.90569 \dots (- 0.69341 \dots)^{4} - 6.93863 \dots (- 0.69341 \dots)^{3} + 3.14215 \dots (- 0.69341 \dots)^{2} + 2.57708 \dots (- 0.69341 \dots) + 8.83297 \dots = 13.71217 \dots

f^{^{'}} (u_{3}) = - 10 (- 0.69341 \dots)^{4} + 43.62278 \dots (- 0.69341 \dots)^{3} - 20.81589 \dots (- 0.69341 \dots)^{2} + 6.28430 \dots (- 0.69341 \dots) + 2.57708 \dots = - 28.64562 \dots

u_{4} = - 0.21473 \dots

Δ u_{4} = ∣ - 0.21473 \dots - (- 0.69341 \dots) ∣ = 0.47868 \dots

Δ u_{4} = 0.47868 \dots

u_{5} = 45.73243 \dots : Δ u_{5} = 45.94716 \dots

f (u_{4}) = - 2 (- 0.21473 \dots)^{5} + 10.90569 \dots (- 0.21473 \dots)^{4} - 6.93863 \dots (- 0.21473 \dots)^{3} + 3.14215 \dots (- 0.21473 \dots)^{2} + 2.57708 \dots (- 0.21473 \dots) + 8.83297 \dots = 8.51727 \dots

f^{^{'}} (u_{4}) = - 10 (- 0.21473 \dots)^{4} + 43.62278 \dots (- 0.21473 \dots)^{3} - 20.81589 \dots (- 0.21473 \dots)^{2} + 6.28430 \dots (- 0.21473 \dots) + 2.57708 \dots = - 0.18537 \dots

u_{5} = 45.73243 \dots

Δ u_{5} = ∣ 45.73243 \dots - (- 0.21473 \dots) ∣ = 45.94716 \dots

Δ u_{5} = 45.94716 \dots

u_{6} = 36.82019 \dots : Δ u_{6} = 8.91223 \dots

f (u_{5}) = - 2 \cdot 45.73243 \dots^{5} + 10.90569 \dots \cdot 45.73243 \dots^{4} - 6.93863 \dots \cdot 45.73243 \dots^{3} + 3.14215 \dots \cdot 45.73243 \dots^{2} + 2.57708 \dots \cdot 45.73243 \dots + 8.83297 \dots = - 353037842.88944 \dots

f^{^{'}} (u_{5}) = - 10 \cdot 45.73243 \dots^{4} + 43.62278 \dots \cdot 45.73243 \dots^{3} - 20.81589 \dots \cdot 45.73243 \dots^{2} + 6.28430 \dots \cdot 45.73243 \dots + 2.57708 \dots = - 39612709.25671 \dots

u_{6} = 36.82019 \dots

Δ u_{6} = ∣ 36.82019 \dots - 45.73243 \dots ∣ = 8.91223 \dots

Δ u_{6} = 8.91223 \dots

u_{7} = 29.69478 \dots : Δ u_{7} = 7.12541 \dots

f (u_{6}) = - 2 \cdot 36.82019 \dots^{5} + 10.90569 \dots \cdot 36.82019 \dots^{4} - 6.93863 \dots \cdot 36.82019 \dots^{3} + 3.14215 \dots \cdot 36.82019 \dots^{2} + 2.57708 \dots \cdot 36.82019 \dots + 8.83297 \dots = - 115648118.63564 \dots

f^{^{'}} (u_{6}) = - 10 \cdot 36.82019 \dots^{4} + 43.62278 \dots \cdot 36.82019 \dots^{3} - 20.81589 \dots \cdot 36.82019 \dots^{2} + 6.28430 \dots \cdot 36.82019 \dots + 2.57708 \dots = - 16230376.60275 \dots

u_{7} = 29.69478 \dots

Δ u_{7} = ∣ 29.69478 \dots - 36.82019 \dots ∣ = 7.12541 \dots

Δ u_{7} = 7.12541 \dots

u_{8} = 24.00013 \dots : Δ u_{8} = 5.69464 \dots

f (u_{7}) = - 2 \cdot 29.69478 \dots^{5} + 10.90569 \dots \cdot 29.69478 \dots^{4} - 6.93863 \dots \cdot 29.69478 \dots^{3} + 3.14215 \dots \cdot 29.69478 \dots^{2} + 2.57708 \dots \cdot 29.69478 \dots + 8.83297 \dots = - 37876817.50021 \dots

f^{^{'}} (u_{7}) = - 10 \cdot 29.69478 \dots^{4} + 43.62278 \dots \cdot 29.69478 \dots^{3} - 20.81589 \dots \cdot 29.69478 \dots^{2} + 6.28430 \dots \cdot 29.69478 \dots + 2.57708 \dots = - 6651300.86677 \dots

u_{8} = 24.00013 \dots

Δ u_{8} = ∣ 24.00013 \dots - 29.69478 \dots ∣ = 5.69464 \dots

Δ u_{8} = 5.69464 \dots

u_{9} = 19.45186 \dots : Δ u_{9} = 4.54827 \dots

f (u_{8}) = - 2 \cdot 24.00013 \dots^{5} + 10.90569 \dots \cdot 24.00013 \dots^{4} - 6.93863 \dots \cdot 24.00013 \dots^{3} + 3.14215 \dots \cdot 24.00013 \dots^{2} + 2.57708 \dots \cdot 24.00013 \dots + 8.83297 \dots = - 12401417.47322 \dots

f^{^{'}} (u_{8}) = - 10 \cdot 24.00013 \dots^{4} + 43.62278 \dots \cdot 24.00013 \dots^{3} - 20.81589 \dots \cdot 24.00013 \dots^{2} + 6.28430 \dots \cdot 24.00013 \dots + 2.57708 \dots = - 2726621.64422 \dots

u_{9} = 19.45186 \dots

Δ u_{9} = ∣ 19.45186 \dots - 24.00013 \dots ∣ = 4.54827 \dots

Δ u_{9} = 4.54827 \dots

u_{10} = 15.82312 \dots : Δ u_{10} = 3.62873 \dots

f (u_{9}) = - 2 \cdot 19.45186 \dots^{5} + 10.90569 \dots \cdot 19.45186 \dots^{4} - 6.93863 \dots \cdot 19.45186 \dots^{3} + 3.14215 \dots \cdot 19.45186 \dots^{2} + 2.57708 \dots \cdot 19.45186 \dots + 8.83297 \dots = - 4058236.53789 \dots

f^{^{'}} (u_{9}) = - 10 \cdot 19.45186 \dots^{4} + 43.62278 \dots \cdot 19.45186 \dots^{3} - 20.81589 \dots \cdot 19.45186 \dots^{2} + 6.28430 \dots \cdot 19.45186 \dots + 2.57708 \dots = - 1118360.52689 \dots

u_{10} = 15.82312 \dots

Δ u_{10} = ∣ 15.82312 \dots - 19.45186 \dots ∣ = 3.62873 \dots

Δ u_{10} = 3.62873 \dots

u_{11} = 12.93345 \dots : Δ u_{11} = 2.88967 \dots

f (u_{10}) = - 2 \cdot 15.82312 \dots^{5} + 10.90569 \dots \cdot 15.82312 \dots^{4} - 6.93863 \dots \cdot 15.82312 \dots^{3} + 3.14215 \dots \cdot 15.82312 \dots^{2} + 2.57708 \dots \cdot 15.82312 \dots + 8.83297 \dots = - 1326791.95496 \dots

f^{^{'}} (u_{10}) = - 10 \cdot 15.82312 \dots^{4} + 43.62278 \dots \cdot 15.82312 \dots^{3} - 20.81589 \dots \cdot 15.82312 \dots^{2} + 6.28430 \dots \cdot 15.82312 \dots + 2.57708 \dots = - 459149.56948 \dots

u_{11} = 12.93345 \dots

Δ u_{11} = ∣ 12.93345 \dots - 15.82312 \dots ∣ = 2.88967 \dots

Δ u_{11} = 2.88967 \dots

u_{12} = 10.64002 \dots : Δ u_{12} = 2.29343 \dots

f (u_{11}) = - 2 \cdot 12.93345 \dots^{5} + 10.90569 \dots \cdot 12.93345 \dots^{4} - 6.93863 \dots \cdot 12.93345 \dots^{3} + 3.14215 \dots \cdot 12.93345 \dots^{2} + 2.57708 \dots \cdot 12.93345 \dots + 8.83297 \dots = - 433068.72392 \dots

f^{^{'}} (u_{11}) = - 10 \cdot 12.93345 \dots^{4} + 43.62278 \dots \cdot 12.93345 \dots^{3} - 20.81589 \dots \cdot 12.93345 \dots^{2} + 6.28430 \dots \cdot 12.93345 \dots + 2.57708 \dots = - 188829.97467 \dots

u_{12} = 10.64002 \dots

Δ u_{12} = ∣ 10.64002 \dots - 12.93345 \dots ∣ = 2.29343 \dots

Δ u_{12} = 2.29343 \dots

u_{13} = 8.83106 \dots : Δ u_{13} = 1.80895 \dots

f (u_{12}) = - 2 \cdot 10.64002 \dots^{5} + 10.90569 \dots \cdot 10.64002 \dots^{4} - 6.93863 \dots \cdot 10.64002 \dots^{3} + 3.14215 \dots \cdot 10.64002 \dots^{2} + 2.57708 \dots \cdot 10.64002 \dots + 8.83297 \dots = - 140929.23683 \dots

f^{^{'}} (u_{12}) = - 10 \cdot 10.64002 \dots^{4} + 43.62278 \dots \cdot 10.64002 \dots^{3} - 20.81589 \dots \cdot 10.64002 \dots^{2} + 6.28430 \dots \cdot 10.64002 \dots + 2.57708 \dots = - 77906.25228 \dots

u_{13} = 8.83106 \dots

Δ u_{13} = ∣ 8.83106 \dots - 10.64002 \dots ∣ = 1.80895 \dots

Δ u_{13} = 1.80895 \dots

u_{14} = 7.42130 \dots : Δ u_{14} = 1.40976 \dots

f (u_{13}) = - 2 \cdot 8.83106 \dots^{5} + 10.90569 \dots \cdot 8.83106 \dots^{4} - 6.93863 \dots \cdot 8.83106 \dots^{3} + 3.14215 \dots \cdot 8.83106 \dots^{2} + 2.57708 \dots \cdot 8.83106 \dots + 8.83297 \dots = - 45595.29435 \dots

f^{^{'}} (u_{13}) = - 10 \cdot 8.83106 \dots^{4} + 43.62278 \dots \cdot 8.83106 \dots^{3} - 20.81589 \dots \cdot 8.83106 \dots^{2} + 6.28430 \dots \cdot 8.83106 \dots + 2.57708 \dots = - 32342.49741 \dots

u_{14} = 7.42130 \dots

Δ u_{14} = ∣ 7.42130 \dots - 8.83106 \dots ∣ = 1.40976 \dots

Δ u_{14} = 1.40976 \dots

u_{15} = 6.34950 \dots : Δ u_{15} = 1.07179 \dots

f (u_{14}) = - 2 \cdot 7.42130 \dots^{5} + 10.90569 \dots \cdot 7.42130 \dots^{4} - 6.93863 \dots \cdot 7.42130 \dots^{3} + 3.14215 \dots \cdot 7.42130 \dots^{2} + 2.57708 \dots \cdot 7.42130 \dots + 8.83297 \dots = - 14576.98569 \dots

f^{^{'}} (u_{14}) = - 10 \cdot 7.42130 \dots^{4} + 43.62278 \dots \cdot 7.42130 \dots^{3} - 20.81589 \dots \cdot 7.42130 \dots^{2} + 6.28430 \dots \cdot 7.42130 \dots + 2.57708 \dots = - 13600.48355 \dots

u_{15} = 6.34950 \dots

Δ u_{15} = ∣ 6.34950 \dots - 7.42130 \dots ∣ = 1.07179 \dots

Δ u_{15} = 1.07179 \dots

u_{16} = 5.57803 \dots : Δ u_{16} = 0.77146 \dots

f (u_{15}) = - 2 \cdot 6.34950 \dots^{5} + 10.90569 \dots \cdot 6.34950 \dots^{4} - 6.93863 \dots \cdot 6.34950 \dots^{3} + 3.14215 \dots \cdot 6.34950 \dots^{2} + 2.57708 \dots \cdot 6.34950 \dots + 8.83297 \dots = - 4539.15945 \dots

f^{^{'}} (u_{15}) = - 10 \cdot 6.34950 \dots^{4} + 43.62278 \dots \cdot 6.34950 \dots^{3} - 20.81589 \dots \cdot 6.34950 \dots^{2} + 6.28430 \dots \cdot 6.34950 \dots + 2.57708 \dots = - 5883.78460 \dots

u_{16} = 5.57803 \dots

Δ u_{16} = ∣ 5.57803 \dots - 6.34950 \dots ∣ = 0.77146 \dots

Δ u_{16} = 0.77146 \dots

u_{17} = 5.09067 \dots : Δ u_{17} = 0.48736 \dots

f (u_{16}) = - 2 \cdot 5.57803 \dots^{5} + 10.90569 \dots \cdot 5.57803 \dots^{4} - 6.93863 \dots \cdot 5.57803 \dots^{3} + 3.14215 \dots \cdot 5.57803 \dots^{2} + 2.57708 \dots \cdot 5.57803 \dots + 8.83297 \dots = - 1325.66062 \dots

f^{^{'}} (u_{16}) = - 10 \cdot 5.57803 \dots^{4} + 43.62278 \dots \cdot 5.57803 \dots^{3} - 20.81589 \dots \cdot 5.57803 \dots^{2} + 6.28430 \dots \cdot 5.57803 \dots + 2.57708 \dots = - 2720.07531 \dots

u_{17} = 5.09067 \dots

Δ u_{17} = ∣ 5.09067 \dots - 5.57803 \dots ∣ = 0.48736 \dots

Δ u_{17} = 0.48736 \dots

u_{18} = 4.86858 \dots : Δ u_{18} = 0.22208 \dots

f (u_{17}) = - 2 \cdot 5.09067 \dots^{5} + 10.90569 \dots \cdot 5.09067 \dots^{4} - 6.93863 \dots \cdot 5.09067 \dots^{3} + 3.14215 \dots \cdot 5.09067 \dots^{2} + 2.57708 \dots \cdot 5.09067 \dots + 8.83297 \dots = - 325.52900 \dots

f^{^{'}} (u_{17}) = - 10 \cdot 5.09067 \dots^{4} + 43.62278 \dots \cdot 5.09067 \dots^{3} - 20.81589 \dots \cdot 5.09067 \dots^{2} + 6.28430 \dots \cdot 5.09067 \dots + 2.57708 \dots = - 1465.80116 \dots

u_{18} = 4.86858 \dots

Δ u_{18} = ∣ 4.86858 \dots - 5.09067 \dots ∣ = 0.22208 \dots

Δ u_{18} = 0.22208 \dots

u_{19} = 4.82230 \dots : Δ u_{19} = 0.04628 \dots

f (u_{18}) = - 2 \cdot 4.86858 \dots^{5} + 10.90569 \dots \cdot 4.86858 \dots^{4} - 6.93863 \dots \cdot 4.86858 \dots^{3} + 3.14215 \dots \cdot 4.86858 \dots^{2} + 2.57708 \dots \cdot 4.86858 \dots + 8.83297 \dots = - 48.34478 \dots

f^{^{'}} (u_{18}) = - 10 \cdot 4.86858 \dots^{4} + 43.62278 \dots \cdot 4.86858 \dots^{3} - 20.81589 \dots \cdot 4.86858 \dots^{2} + 6.28430 \dots \cdot 4.86858 \dots + 2.57708 \dots = - 1044.51538 \dots

u_{19} = 4.82230 \dots

Δ u_{19} = ∣ 4.82230 \dots - 4.86858 \dots ∣ = 0.04628 \dots

Δ u_{19} = 0.04628 \dots

u_{20} = 4.82043 \dots : Δ u_{20} = 0.00186 \dots

f (u_{19}) = - 2 \cdot 4.82230 \dots^{5} + 10.90569 \dots \cdot 4.82230 \dots^{4} - 6.93863 \dots \cdot 4.82230 \dots^{3} + 3.14215 \dots \cdot 4.82230 \dots^{2} + 2.57708 \dots \cdot 4.82230 \dots + 8.83297 \dots = - 1.80563 \dots

f^{^{'}} (u_{19}) = - 10 \cdot 4.82230 \dots^{4} + 43.62278 \dots \cdot 4.82230 \dots^{3} - 20.81589 \dots \cdot 4.82230 \dots^{2} + 6.28430 \dots \cdot 4.82230 \dots + 2.57708 \dots = - 967.05976 \dots

u_{20} = 4.82043 \dots

Δ u_{20} = ∣ 4.82043 \dots - 4.82230 \dots ∣ = 0.00186 \dots

Δ u_{20} = 0.00186 \dots

u_{21} = 4.82043 \dots : Δ u_{21} = 2.95792 E - 6

f (u_{20}) = - 2 \cdot 4.82043 \dots^{5} + 10.90569 \dots \cdot 4.82043 \dots^{4} - 6.93863 \dots \cdot 4.82043 \dots^{3} + 3.14215 \dots \cdot 4.82043 \dots^{2} + 2.57708 \dots \cdot 4.82043 \dots + 8.83297 \dots = - 0.00285 \dots

f^{^{'}} (u_{20}) = - 10 \cdot 4.82043 \dots^{4} + 43.62278 \dots \cdot 4.82043 \dots^{3} - 20.81589 \dots \cdot 4.82043 \dots^{2} + 6.28430 \dots \cdot 4.82043 \dots + 2.57708 \dots = - 964.00633 \dots

u_{21} = 4.82043 \dots

Δ u_{21} = ∣ 4.82043 \dots - 4.82043 \dots ∣ = 2.95792 E - 6

Δ u_{21} = 2.95792 E - 6

u_{22} = 4.82043 \dots : Δ u_{22} = 7.4149 E - 12

f (u_{21}) = - 2 \cdot 4.82043 \dots^{5} + 10.90569 \dots \cdot 4.82043 \dots^{4} - 6.93863 \dots \cdot 4.82043 \dots^{3} + 3.14215 \dots \cdot 4.82043 \dots^{2} + 2.57708 \dots \cdot 4.82043 \dots + 8.83297 \dots = - 7.14797 E - 9

f^{^{'}} (u_{21}) = - 10 \cdot 4.82043 \dots^{4} + 43.62278 \dots \cdot 4.82043 \dots^{3} - 20.81589 \dots \cdot 4.82043 \dots^{2} + 6.28430 \dots \cdot 4.82043 \dots + 2.57708 \dots = - 964.00149 \dots

u_{22} = 4.82043 \dots

Δ u_{22} = ∣ 4.82043 \dots - 4.82043 \dots ∣ = 7.4149 E - 12

Δ u_{22} = 7.4149 E - 12

u \approx 4.82043 \dots

Apply long division:

\frac{- 2 u ^{5} + 10.90569 \dots u ^{4} - 6.93863 \dots u ^{3} + 3.14215 \dots u ^{2} + 2.57708 \dots u + 8.83297 \dots}{u - 4.82043 \dots} = - 2 u^{4} + 1.26482 \dots u^{3} - 0.84160 \dots u^{2} - 0.91474 \dots u - 1.83240 \dots

- 2 u^{4} + 1.26482 \dots u^{3} - 0.84160 \dots u^{2} - 0.91474 \dots u - 1.83240 \dots \approx 0

Find one solution for

- 2 u^{4} + 1.26482 \dots u^{3} - 0.84160 \dots u^{2} - 0.91474 \dots u - 1.83240 \dots = 0

using Newton-Raphson:

No Solution for

u \in R

- 2 u^{4} + 1.26482 \dots u^{3} - 0.84160 \dots u^{2} - 0.91474 \dots u - 1.83240 \dots = 0

Newton-Raphson Approximation Definition

f (u) = - 2 u^{4} + 1.26482 \dots u^{3} - 0.84160 \dots u^{2} - 0.91474 \dots u - 1.83240 \dots

Find

f^{^{'}} (u) : - 8 u^{3} + 3.79448 \dots u^{2} - 1.68320 \dots u - 0.91474 \dots

\frac{d}{d u} (- 2 u^{4} + 1.26482 \dots u^{3} - 0.84160 \dots u^{2} - 0.91474 \dots u - 1.83240 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d u} (2 u^{4}) + \frac{d}{d u} (1.26482 \dots u^{3}) - \frac{d}{d u} (0.84160 \dots u^{2}) - \frac{d}{d u} (0.91474 \dots u) - \frac{d}{d u} (1.83240 \dots)

\frac{d}{d u} (2 u^{4}) = 8 u^{3}

\frac{d}{d u} (2 u^{4})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{4})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 4 u^{4 - 1}

Simplify

= 8 u^{3}

\frac{d}{d u} (1.26482 \dots u^{3}) = 3.79448 \dots u^{2}

\frac{d}{d u} (1.26482 \dots u^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 1.26482 \dots \frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 1.26482 \dots \cdot 3 u^{3 - 1}

Simplify

= 3.79448 \dots u^{2}

\frac{d}{d u} (0.84160 \dots u^{2}) = 1.68320 \dots u

\frac{d}{d u} (0.84160 \dots u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 0.84160 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 0.84160 \dots \cdot 2 u^{2 - 1}

Simplify

= 1.68320 \dots u

\frac{d}{d u} (0.91474 \dots u) = 0.91474 \dots

\frac{d}{d u} (0.91474 \dots u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 0.91474 \dots \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 0.91474 \dots \cdot 1

Simplify

= 0.91474 \dots

\frac{d}{d u} (1.83240 \dots) = 0

\frac{d}{d u} (1.83240 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= - 8 u^{3} + 3.79448 \dots u^{2} - 1.68320 \dots u - 0.91474 \dots - 0

Simplify

= - 8 u^{3} + 3.79448 \dots u^{2} - 1.68320 \dots u - 0.91474 \dots

Let

u_{0} = - 2

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 1.44275 \dots : Δ u_{1} = 0.55724 \dots

f (u_{0}) = - 2 (- 2)^{4} + 1.26482 \dots (- 2)^{3} - 0.84160 \dots (- 2)^{2} - 0.91474 \dots (- 2) - 1.83240 \dots = - 45.48795 \dots

f^{^{'}} (u_{0}) = - 8 (- 2)^{3} + 3.79448 \dots (- 2)^{2} - 1.68320 \dots (- 2) - 0.91474 \dots = 81.62962 \dots

u_{1} = - 1.44275 \dots

Δ u_{1} = ∣ - 1.44275 \dots - (- 2) ∣ = 0.55724 \dots

Δ u_{1} = 0.55724 \dots

u_{2} = - 1.00226 \dots : Δ u_{2} = 0.44048 \dots

f (u_{1}) = - 2 (- 1.44275 \dots)^{4} + 1.26482 \dots (- 1.44275 \dots)^{3} - 0.84160 \dots (- 1.44275 \dots)^{2} - 0.91474 \dots (- 1.44275 \dots) - 1.83240 \dots = - 14.72848 \dots

f^{^{'}} (u_{1}) = - 8 (- 1.44275 \dots)^{3} + 3.79448 \dots (- 1.44275 \dots)^{2} - 1.68320 \dots (- 1.44275 \dots) - 0.91474 \dots = 33.43713 \dots

u_{2} = - 1.00226 \dots

Δ u_{2} = ∣ - 1.00226 \dots - (- 1.44275 \dots) ∣ = 0.44048 \dots

Δ u_{2} = 0.44048 \dots

u_{3} = - 0.60248 \dots : Δ u_{3} = 0.39978 \dots

f (u_{2}) = - 2 (- 1.00226 \dots)^{4} + 1.26482 \dots (- 1.00226 \dots)^{3} - 0.84160 \dots (- 1.00226 \dots)^{2} - 0.91474 \dots (- 1.00226 \dots) - 1.83240 \dots = - 5.05267 \dots

f^{^{'}} (u_{2}) = - 8 (- 1.00226 \dots)^{3} + 3.79448 \dots (- 1.00226 \dots)^{2} - 1.68320 \dots (- 1.00226 \dots) - 0.91474 \dots = 12.63858 \dots

u_{3} = - 0.60248 \dots

Δ u_{3} = ∣ - 0.60248 \dots - (- 1.00226 \dots) ∣ = 0.39978 \dots

Δ u_{3} = 0.39978 \dots

u_{4} = 0.05675 \dots : Δ u_{4} = 0.65924 \dots

f (u_{3}) = - 2 (- 0.60248 \dots)^{4} + 1.26482 \dots (- 0.60248 \dots)^{3} - 0.84160 \dots (- 0.60248 \dots)^{2} - 0.91474 \dots (- 0.60248 \dots) - 1.83240 \dots = - 2.12691 \dots

f^{^{'}} (u_{3}) = - 8 (- 0.60248 \dots)^{3} + 3.79448 \dots (- 0.60248 \dots)^{2} - 1.68320 \dots (- 0.60248 \dots) - 0.91474 \dots = 3.22630 \dots

u_{4} = 0.05675 \dots

Δ u_{4} = ∣ 0.05675 \dots - (- 0.60248 \dots) ∣ = 0.65924 \dots

Δ u_{4} = 0.65924 \dots

u_{5} = - 1.83097 \dots : Δ u_{5} = 1.88772 \dots

f (u_{4}) = - 2 \cdot 0.05675 \dots^{4} + 1.26482 \dots \cdot 0.05675 \dots^{3} - 0.84160 \dots \cdot 0.05675 \dots^{2} - 0.91474 \dots \cdot 0.05675 \dots - 1.83240 \dots = - 1.88681 \dots

f^{^{'}} (u_{4}) = - 8 \cdot 0.05675 \dots^{3} + 3.79448 \dots \cdot 0.05675 \dots^{2} - 1.68320 \dots \cdot 0.05675 \dots - 0.91474 \dots = - 0.99951 \dots

u_{5} = - 1.83097 \dots

Δ u_{5} = ∣ - 1.83097 \dots - 0.05675 \dots ∣ = 1.88772 \dots

Δ u_{5} = 1.88772 \dots

u_{6} = - 1.31185 \dots : Δ u_{6} = 0.51912 \dots

f (u_{5}) = - 2 (- 1.83097 \dots)^{4} + 1.26482 \dots (- 1.83097 \dots)^{3} - 0.84160 \dots (- 1.83097 \dots)^{2} - 0.91474 \dots (- 1.83097 \dots) - 1.83240 \dots = - 33.22099 \dots

f^{^{'}} (u_{5}) = - 8 (- 1.83097 \dots)^{3} + 3.79448 \dots (- 1.83097 \dots)^{2} - 1.68320 \dots (- 1.83097 \dots) - 0.91474 \dots = 63.99442 \dots

u_{6} = - 1.31185 \dots

Δ u_{6} = ∣ - 1.31185 \dots - (- 1.83097 \dots) ∣ = 0.51912 \dots

Δ u_{6} = 0.51912 \dots

u_{7} = - 0.89231 \dots : Δ u_{7} = 0.41954 \dots

f (u_{6}) = - 2 (- 1.31185 \dots)^{4} + 1.26482 \dots (- 1.31185 \dots)^{3} - 0.84160 \dots (- 1.31185 \dots)^{2} - 0.91474 \dots (- 1.31185 \dots) - 1.83240 \dots = - 10.85966 \dots

f^{^{'}} (u_{6}) = - 8 (- 1.31185 \dots)^{3} + 3.79448 \dots (- 1.31185 \dots)^{2} - 1.68320 \dots (- 1.31185 \dots) - 0.91474 \dots = 25.88464 \dots

u_{7} = - 0.89231 \dots

Δ u_{7} = ∣ - 0.89231 \dots - (- 1.31185 \dots) ∣ = 0.41954 \dots

Δ u_{7} = 0.41954 \dots

u_{8} = - 0.47768 \dots : Δ u_{8} = 0.41462 \dots

f (u_{7}) = - 2 (- 0.89231 \dots)^{4} + 1.26482 \dots (- 0.89231 \dots)^{3} - 0.84160 \dots (- 0.89231 \dots)^{2} - 0.91474 \dots (- 0.89231 \dots) - 1.83240 \dots = - 3.85282 \dots

f^{^{'}} (u_{7}) = - 8 (- 0.89231 \dots)^{3} + 3.79448 \dots (- 0.89231 \dots)^{2} - 1.68320 \dots (- 0.89231 \dots) - 0.91474 \dots = 9.29225 \dots

u_{8} = - 0.47768 \dots

Δ u_{8} = ∣ - 0.47768 \dots - (- 0.89231 \dots) ∣ = 0.41462 \dots

Δ u_{8} = 0.41462 \dots

u_{9} = 0.64668 \dots : Δ u_{9} = 1.12436 \dots

f (u_{8}) = - 2 (- 0.47768 \dots)^{4} + 1.26482 \dots (- 0.47768 \dots)^{3} - 0.84160 \dots (- 0.47768 \dots)^{2} - 0.91474 \dots (- 0.47768 \dots) - 1.83240 \dots = - 1.82947 \dots

f^{^{'}} (u_{8}) = - 8 (- 0.47768 \dots)^{3} + 3.79448 \dots (- 0.47768 \dots)^{2} - 1.68320 \dots (- 0.47768 \dots) - 0.91474 \dots = 1.62711 \dots

u_{9} = 0.64668 \dots

Δ u_{9} = ∣ 0.64668 \dots - (- 0.47768 \dots) ∣ = 1.12436 \dots

Δ u_{9} = 1.12436 \dots

u_{10} = - 0.43226 \dots : Δ u_{10} = 1.07894 \dots

f (u_{9}) = - 2 \cdot 0.64668 \dots^{4} + 1.26482 \dots \cdot 0.64668 \dots^{3} - 0.84160 \dots \cdot 0.64668 \dots^{2} - 0.91474 \dots \cdot 0.64668 \dots - 1.83240 \dots = - 2.78363 \dots

f^{^{'}} (u_{9}) = - 8 \cdot 0.64668 \dots^{3} + 3.79448 \dots \cdot 0.64668 \dots^{2} - 1.68320 \dots \cdot 0.64668 \dots - 0.91474 \dots = - 2.57995 \dots

u_{10} = - 0.43226 \dots

Δ u_{10} = ∣ - 0.43226 \dots - 0.64668 \dots ∣ = 1.07894 \dots

Δ u_{10} = 1.07894 \dots

u_{11} = 1.07993 \dots : Δ u_{11} = 1.51219 \dots

f (u_{10}) = - 2 (- 0.43226 \dots)^{4} + 1.26482 \dots (- 0.43226 \dots)^{3} - 0.84160 \dots (- 0.43226 \dots)^{2} - 0.91474 \dots (- 0.43226 \dots) - 1.83240 \dots = - 1.76622 \dots

f^{^{'}} (u_{10}) = - 8 (- 0.43226 \dots)^{3} + 3.79448 \dots (- 0.43226 \dots)^{2} - 1.68320 \dots (- 0.43226 \dots) - 0.91474 \dots = 1.16798 \dots

u_{11} = 1.07993 \dots

Δ u_{11} = ∣ 1.07993 \dots - (- 0.43226 \dots) ∣ = 1.51219 \dots

Δ u_{11} = 1.51219 \dots

Cannot find solution

The solutions are

u \approx - 0.45284 \dots, u \approx 4.82043 \dots

u \approx - 0.45284 \dots, u \approx 4.82043 \dots

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

3 \frac{u ^{2} - u ^{- 2}}{u ^{2} + u ^{- 2}}

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

Take the denominator(s) of

5 \frac{2}{u + u ^{- 1}} + 1

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u \approx - 0.45284 \dots, u \approx 4.82043 \dots

u \approx - 0.45284 \dots, u \approx 4.82043 \dots

Substitute back

u = e^{θ},

solve for

θ

Solve

e^{θ} = - 0.45284 \dots :

No Solution for

θ \in R

e^{θ} = - 0.45284 \dots

a^{f (θ)}

cannot be zero or negative for

θ \in R

No Solution for θ \in R

Solve

e^{θ} = 4.82043 \dots : θ = ln (4.82043 \dots)

e^{θ} = 4.82043 \dots

Apply exponent rules

e^{θ} = 4.82043 \dots

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{θ}) = ln (4.82043 \dots)

Apply log rule:

ln (e^{a}) = a

ln (e^{θ}) = θ

θ = ln (4.82043 \dots)

θ = ln (4.82043 \dots)

θ = ln (4.82043 \dots)

θ = ln (4.82043 \dots)

Popular Examples

2cos^2(θ)+cos(θ)=1

2 cos^{2} (θ) + cos (θ) = 1

tan(x)sin(x)-sin(x)=0

tan (x) sin (x) - sin (x) = 0

2sin(x)=csc(x)

2 sin (x) = csc (x)

1+tan(x)=sec(x)

1 + tan (x) = sec (x)

sin(2x)=cos(60)

sin (2 x) = cos (6 0^{\circ})

Frequently Asked Questions (FAQ)

What is the general solution for 3tanh(2θ)=5sech(θ)+1 ?
The general solution for 3tanh(2θ)=5sech(θ)+1 is θ=ln(4.82043…)