What is the tangent of y=(\sqrt[4]{81+x}-3)/x ?

The tangent of y=(\sqrt[4]{81+x}-3)/x is

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tangent of y=(\sqrt[4]{81+x}-3)/x

tangent of

y = \frac{4 81 + x - 3}{x}

y = \frac{a _{0} - 4 ( 4 81 + a _{0} - 3 ) ( a _{0} + 81 ) ^{\frac{3}{4}}}{4 a _{0}^{2} ( a _{0} + 81 ) ^{\frac{3}{4}}} x + \frac{8 ( a _{0} + 81 ) ^{\frac{3}{4}} 4 a _{0} + 81 - 24 ( a _{0} + 81 ) ^{\frac{3}{4}} - a _{0}}{4 a _{0} ( a _{0} + 81 ) ^{\frac{3}{4}}}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, \frac{4 81 + a _{0} - 3}{a _{0}})

Find the slope of

y = \frac{4 81 + x - 3}{x} : \frac{d y}{d x} = \frac{x - 4 ( 4 81 + x - 3 ) ( x + 81 ) ^{\frac{3}{4}}}{4 x ^{2} ( x + 81 ) ^{\frac{3}{4}}}

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{a _{0} - 4 ( 4 81 + a _{0} - 3 ) ( a _{0} + 81 ) ^{\frac{3}{4}}}{4 a _{0}^{2} ( a _{0} + 81 ) ^{\frac{3}{4}}}

Find the line with slope m=

\frac{a _{0} - 4 ( 4 81 + a _{0} - 3 ) ( a _{0} + 81 ) ^{\frac{3}{4}}}{4 a _{0}^{2} ( a _{0} + 81 ) ^{\frac{3}{4}}}

and passing through

(a_{0}, \frac{4 81 + a _{0} - 3}{a _{0}}) : y = \frac{a _{0} - 4 ( 4 81 + a _{0} - 3 ) ( a _{0} + 81 ) ^{\frac{3}{4}}}{4 a _{0}^{2} ( a _{0} + 81 ) ^{\frac{3}{4}}} x + \frac{8 ( a _{0} + 81 ) ^{\frac{3}{4}} 4 a _{0} + 81 - 24 ( a _{0} + 81 ) ^{\frac{3}{4}} - a _{0}}{4 a _{0} ( a _{0} + 81 ) ^{\frac{3}{4}}}

y = \frac{a _{0} - 4 ( 4 81 + a _{0} - 3 ) ( a _{0} + 81 ) ^{\frac{3}{4}}}{4 a _{0}^{2} ( a _{0} + 81 ) ^{\frac{3}{4}}} x + \frac{8 ( a _{0} + 81 ) ^{\frac{3}{4}} 4 a _{0} + 81 - 24 ( a _{0} + 81 ) ^{\frac{3}{4}} - a _{0}}{4 a _{0} ( a _{0} + 81 ) ^{\frac{3}{4}}}

y^{^{'}} + 8 y + 6 x = 0

y = 4 x - 9

\int \frac{( x + 6 )}{x} d x

\frac{d}{d x} (x cos (x) + x^{2} sin (x))

f (x) = 533.33 + (\frac{x}{250})^{2}

What is the tangent of y=(\sqrt[4]{81+x}-3)/x ?
The tangent of y=(\sqrt[4]{81+x}-3)/x is