Question

Differentiation II help - find dy/dx when y=arcsin(e^2x). Will award medal and fan

Answer 1

\[y=\sin^{-1}\left(e^{2x}\right)~~\iff~~\sin y=e^{2x}\]
Implicit differentiation gives
\[\begin{align*}\cos y \frac{dy}{dx}=2e^{2x}~~\implies~~\frac{dy}{dx}&=2e^{2x}\sec y\\
&=2e^{2x}\sec\left(\sin^{-1}\left(e^{2x}\right)\right)
\end{align*}\]

Answer 2

Can you explain the implicit differentiation part? I'm not familiar with it :S

Answer 3

Implicit differentiation is just the chain rule. So if I ask you what the derivative of y^2 is, you would say it's \[\LARGE \frac{d}{dx}(y^2)=2y*\frac{dy}{dx}\] since it's just the derivative of the outside times the derivative of the inside like the chain rule normally says.

Answer 4

Thank you both! I understand :)