Find the derivative of the following function using First Principles.

Question

f(x)= x^2+5^x

Answer 1

the x^2 part is easy, right?

Now note that 5^x = (e^ln5)^x = e^(ln5 x)
Use google and you can find many sites which prove that the derivative of e^u = e^u using first principles. They all use some variation of the fact that

lim(n->0) (1 + 1/n)^n = e