How would I find the derivative of $\displaystyle tan(x)^{secx}$
use parentheses! you have the idea, but the answer as written is wrong.
you should have: $\displaystyle \left( \sec x\tan x \ln ( \tan x) + \sec x \frac{1}{\tan x}\sec^2x \right)e^{\sec x \ln ( \tan x)}$
by the way, this can be simplified a lot, for instance, instead of writing $\displaystyle \sec x \frac 1{\tan x} \sec^2 x$ you could write $\displaystyle \frac {\sec^3 x}{\tan x}$ and you can change the $\displaystyle e^{\sec x \ln \tan x}$ back to it's original form