Question

help with derivatives of inverse functions

Answer 1

\[find f'(x): f(x)=\tan^{-1} (x^4)+(tanx^4)^{-1}+(\tan^{-1} x)^4\]

Answer 2

derive each one piece by piece.

Answer 3

\(\Large\color{black}{ y=\tan^{-1}(x^4) }\)
\(\Large\color{black}{ x=\tan(\sqrt[4]{y}) }\)
chain rule twice, once for the inner functin of tangent, and another one for y prime.

So dy/dx is:
\(\Large\color{black}{ 1=\sec^2(\sqrt[4]{y})\times (\frac{1}{4\sqrt[4]{y^3}})\times y' }\)

Answer 4

\(\Large\color{black}{ 1=[\tan^2(\sqrt[4]{y})+1]\times (\frac{1}{4\sqrt[4]{y^3}})\times y' }\)

Answer 5

I won't do the entire problem, but you get the idea.....

Answer 6

thank you!