Question

How do you differentiate `f(x)=(2x^3)(ln3x)(sinx)` using the product rule?

Answer 1

`f'(x) = 6x^2 \ ln3x \ sinx + 2x^2 \ sinx + 2x^3 \ ln3x \ cosx`

Explanation:

We have:

`f(x)=(2x^3)(ln3x)(sinx)`

We need to use the triple product rule, giving

`f'(x) = (d/dx2x^3)(ln3x)(sinx) + (2x^3)(d/dxln3x)(sinx) + (2x^3)(ln3x)(d/dxsinx)`

`" " = (6x^2)(ln3x)(sinx) + (2x^3)(1/(3x)d/dx3x)(sinx) + (2x^3)(ln3x)(cosx)`

`" " = (6x^2)(ln3x)(sinx) + (2x^3)((3)/(3x))(sinx) + (2x^3)(ln3x)(cosx)`

`" " = 6x^2 \ ln3x \ sinx + 2x^2 \ sinx + 2x^3 \ ln3x \ cosx`