What is the tangent of x^{3/2}+x^{1/2} ?

The tangent of x^{3/2}+x^{1/2} is

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tangent of x^{3/2}+x^{1/2}

tangent of

x^{\frac{3}{2}} + x^{\frac{1}{2}}

y = (\frac{3 a _{0}^{\frac{1}{2}}}{2} + \frac{1}{2 a _{0}^{\frac{1}{2}}}) x + a_{0}^{\frac{3}{2}} + a_{0}^{\frac{1}{2}} + \frac{- 3 a _{0}^{\frac{3}{2}} - a _{0}^{\frac{1}{2}}}{2}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, a_{0}^{\frac{3}{2}} + a_{0}^{\frac{1}{2}})

Find the slope of

y = x^{\frac{3}{2}} + x^{\frac{1}{2}} : \frac{d y}{d x} = \frac{3 x ^{\frac{1}{2}}}{2} + \frac{1}{2 x ^{\frac{1}{2}}}

Compute the slope of

y = x^{\frac{3}{2}} + x^{\frac{1}{2}}

x = a_{0} : m = \frac{3 a _{0}^{\frac{1}{2}}}{2} + \frac{1}{2 a _{0}^{\frac{1}{2}}}

Find the line with slope m=

\frac{3 a _{0}^{\frac{1}{2}}}{2} + \frac{1}{2 a _{0}^{\frac{1}{2}}}

and passing through

(a_{0}, a_{0}^{\frac{3}{2}} + a_{0}^{\frac{1}{2}}) : y = (\frac{3 a _{0}^{\frac{1}{2}}}{2} + \frac{1}{2 a _{0}^{\frac{1}{2}}}) x + a_{0}^{\frac{3}{2}} + a_{0}^{\frac{1}{2}} + \frac{- 3 a _{0}^{\frac{3}{2}} - a _{0}^{\frac{1}{2}}}{2}

y = (\frac{3 a _{0}^{\frac{1}{2}}}{2} + \frac{1}{2 a _{0}^{\frac{1}{2}}}) x + a_{0}^{\frac{3}{2}} + a_{0}^{\frac{1}{2}} + \frac{- 3 a _{0}^{\frac{3}{2}} - a _{0}^{\frac{1}{2}}}{2}

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