Calculus: Puzzling Question?

Question

Calculate f'' for f(x)=g(e^2x) where g is a function defined for all real numbers and g admits second order derivative

I don't even know where to start with this question, so any tips or help will be greatly appreciated.

thanks. and to find the second derivative (f'')?

Answer 1

First use the chain rule on f(x)=g(e^2x) to get
f '(x)=g '(x)(2e^2x)
now use the product rule on each factor in parenthesis
f ''(x)=g ''(e^2x)(2e^2x)(2e^2x)+g '(e^2x)(4e^2x)
=g ''(e^2x)(4e^4x)+g '(e^2x)(4e^2x)

Answer 2

First, use the product rule:

d / dx ( g ) ( e^2x ) = ( g ) d / dx ( e^2x ) + ( e^2x ) d / dx ( g )

The derivative of g is just g'
The derivative of e^2x is 2 e^2x

Answer 3

Chain rule question:
f'(x) = 2e^(2x)g'(e^2x).

Answer 4

Chain Rule:

h(x) = e^2x
g(x) = (blah blah)
f(x) = g(h(x))