What is the tangent of 9x^{1/2}+x^{3/2} ?

The tangent of 9x^{1/2}+x^{3/2} is

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tangent of 9x^{1/2}+x^{3/2}

tangent of

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

y = (\frac{9}{2 a _{0}^{\frac{1}{2}}} + \frac{3 a _{0}^{\frac{1}{2}}}{2}) x + 9 a_{0}^{\frac{1}{2}} + a_{0}^{\frac{3}{2}} + \frac{- 9 a _{0}^{\frac{1}{2}} - 3 a _{0}^{\frac{3}{2}}}{2}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, 9 a_{0}^{\frac{1}{2}} + a_{0}^{\frac{3}{2}})

Find the slope of

y = 9 x^{\frac{1}{2}} + x^{\frac{3}{2}} : \frac{d y}{d x} = \frac{9}{2 x ^{\frac{1}{2}}} + \frac{3 x ^{\frac{1}{2}}}{2}

Compute the slope of

y = 9 x^{\frac{1}{2}} + x^{\frac{3}{2}}

x = a_{0} : m = \frac{9}{2 a _{0}^{\frac{1}{2}}} + \frac{3 a _{0}^{\frac{1}{2}}}{2}

Find the line with slope m=

\frac{9}{2 a _{0}^{\frac{1}{2}}} + \frac{3 a _{0}^{\frac{1}{2}}}{2}

and passing through

(a_{0}, 9 a_{0}^{\frac{1}{2}} + a_{0}^{\frac{3}{2}}) : y = (\frac{9}{2 a _{0}^{\frac{1}{2}}} + \frac{3 a _{0}^{\frac{1}{2}}}{2}) x + 9 a_{0}^{\frac{1}{2}} + a_{0}^{\frac{3}{2}} + \frac{- 9 a _{0}^{\frac{1}{2}} - 3 a _{0}^{\frac{3}{2}}}{2}

y = (\frac{9}{2 a _{0}^{\frac{1}{2}}} + \frac{3 a _{0}^{\frac{1}{2}}}{2}) x + 9 a_{0}^{\frac{1}{2}} + a_{0}^{\frac{3}{2}} + \frac{- 9 a _{0}^{\frac{1}{2}} - 3 a _{0}^{\frac{3}{2}}}{2}

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