What is (\partial)/(\partial x)(y/(x(x+y))) ?

The answer to (\partial)/(\partial x)(y/(x(x+y))) is -(y(2x+y))/(x^2(x+y)^2)

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(\partial)/(\partial x)(y/(x(x+y)))

\frac{\partial}{\partial x} (\frac{y}{x ( x + y )})

- \frac{y ( 2 x + y )}{x ^{2} ( x + y ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{y}{x ( x + y )})

Treat

y

as a constant

Take the constant out:

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

= y \frac{\partial}{\partial x} (\frac{1}{x ( x + y )})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= y \frac{\partial}{\partial x} ((x (x + y))^{- 1})

Apply the chain rule:

- \frac{1}{( x ( x + y ) ) ^{2}} \frac{\partial}{\partial x} (x (x + y))

= - \frac{1}{( x ( x + y ) ) ^{2}} \frac{\partial}{\partial x} (x (x + y))

\frac{\partial}{\partial x} (x (x + y)) = 2 x + y

= y (- \frac{1}{( x ( x + y ) ) ^{2}} (2 x + y))

Simplify

y (- \frac{1}{( x ( x + y ) ) ^{2}} (2 x + y)) : - \frac{y ( 2 x + y )}{x ^{2} ( x + y ) ^{2}}

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

x \to 7 lim (\frac{x + 8}{x - 7})

\int - \frac{1}{\frac{2}{x} y} d y

sin (tan (5 x))

\frac{d}{d x} (- 2 (e^{- x^{2}} - 2 e^{- x^{2}} x^{2}))

\frac{\partial}{\partial x} (x y + x z - 2 yz)

What is (\partial)/(\partial x)(y/(x(x+y))) ?
The answer to (\partial)/(\partial x)(y/(x(x+y))) is -(y(2x+y))/(x^2(x+y)^2)