Question

How would one take the derivative of "f(x)^[g(x)]? Is there a formula I can follow to differentiate functions like that?

Answer 1

you will have to take the natural log of both sides.
I will illustrate with an example

e.g.

y = (x^2+1)^(2x+3)
ln y = ln ((x^2+1)^(2x+3))
ln y = (2x+3)(ln(x^2+1)

now differentiate, using the product rule on the right side.
(dy/dx)/y = (2x+3)(2x/(x^2+1)) + 2ln(x^2+1)
dy/dx = y((2x+3)(2x/(x^2+1)) + 2ln(x^2+1))
you could then replace y with its original value and simplify, but that would be a general method.