Find the derivative of $\displaystyle tan^{-1}(x^2)$.
Could someone explain how to solve this?
Chain rule. Let $\displaystyle u = x^2 $ so $\displaystyle y = \tan^{-1} u$. So
$\displaystyle \frac{dy}{dx} = \frac{dy}{du}\, \cdot \, \frac{du}{dx}$ and calculating each derivative gives $\displaystyle \frac{dy}{du} = \frac{1}{1+u^2}$ and $\displaystyle \frac{du}{dx} = 2x$.
Substituting gives
$\displaystyle \frac{dy}{dx} = \frac{1}{1+u^2}\, \cdot2x = \frac{2x}{1+x^4}.$