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}

EN : T i tl e G e n er a l Eq u a t i o n Sl o p e A t P o in t 2 Eq : 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}

(4, - 8), (8, 5)

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