I was having trouble with the following question about finding the derivative of:
y=(tan^-1(x/2))^4
-i utilized the chain rule but this came out to be wrong.
$\displaystyle \displaystyle \begin{align*} y &= \left[\arctan{\left(\frac{x}{2}\right)}\right]^4 \\ y^{\frac{1}{4}} &= \arctan{\left(\frac{x}{2}\right)} \\ \tan{\left(y^{\frac{1}{4}}\right)} &= \frac{x}{2} \\ \frac{d}{dx}\left[\tan{\left(y^{\frac{1}{4}}\right)}\right] &= \frac{d}{dx}\left(\frac{x}{2}\right) \\ \frac{d}{dy}\left[\tan{\left(y^{\frac{1}{4}}\right)}\right]\,\frac{dy}{dx} &= \frac{1}{2} \\ \frac{1}{4}y^{-\frac{3}{4}}\sec^2{\left(y^{\frac{1}{4}}\right)}\, \frac{dy}{dx} &= \frac{1}{2} \\ \frac{dy}{dx} &= 2y^{\frac{3}{4}}\cos^2{\left(y^{\frac{1}{4}}\right )}\end{align*}$
You could convert it back to a function of $\displaystyle \displaystyle x$ now if you want.