How do you take the derivative of f(x) = arctan(2^x)?
well, all you have to know is this:
$\displaystyle \frac{dy}{dx}(arctan(x)) = \frac{1}{1 + x^2}$
Then, you have to apply the chain rule to find the derivative of the inside function $\displaystyle x^2$.
So, here is your entire problem.
$\displaystyle f(x) = arctan(x^2)$
$\displaystyle f'(x) = \frac{1}{1+(x^2)^2}*2x$
$\displaystyle f'(x) = \frac{2x}{1+x^4} $