Find the derivative of f(x) = (e^3x)^4 two different ways, and the derivative of g(x) = ln (x^3) two different ways.

Answer 1

f(x) = (e^3x)^4

Method 1: simplify, then take the derivative:

f(x) = (e^3x)^4 = e^12x

use the chain rule: u = 12x, f(x) = e^u

f'(x) = df/du * du/dx = e^u * 12

f'(x) = 12e^12x

Method 2: apply the chain rule directly:

u = e^3x, f(x) = (u)^4

f'(x) = df/du * du/dx = 4u^3 * d/dx(e^3x)

again using the chain rule, d/dx(e^3x) = 3e^3x

SO:

f'(x) = 4u^3 * 3(e^3x)

but u = e^3x, so f'(x) = 4(e^3x)^3 * 3(e^3x)

f'(x) = 12 * (e^3x)^4

f'(x) = 12e^12x