Question

What is the derivative of `f(x)=2lnx^3`?

Answer 1

Using the chain rule, we can rename `u=x^3`, and thus, start working with `f(x)=f(u)=2ln(u)`.

The chain rule states that

`(dy)/(dx)=(dy)/(du)(du)/(dx)`

So,

`(dy)/(du)=2*1/u=2/u`

`(du)/(dx)=3x^2`

Thus,

`(dy)/(dx)=2/u*3x^2=(6x^2)/u=(6cancel(x^2))/x^cancel(3)=color(green)(6/x)`

Answer 2

Alternative solution:

`f(x) = 2lnx^3=6lnx`

So,

`f'(x) = 6 * 1/x = 6/x`