Find the derivative of f(x)

Question

Find f'(x) for f(x) = ln(x³ + e^4x).

Answer 1

f(x) = ln(x³ + e^4x)

To find the 1st derivative, we will use the chain rule---twice!

Start by defining u(x) = (x³ + e^4x).

The chain rule: given f(u), where u is a function of x, f'(u) - df/du * du/dx

So: f = ln(u), and f' = 1/u) * du/dx

We will also use the chain rule to find du/dx:

du/dx = 3x² + e^4x * 4

Putting it all together:

f' = (3x² + 4e^4x ) / (x³ + e^4x)