Question

How do you find the derivative of `f(x) = 2x^3 e^(9x)`?

Answer 1

The answer is : `f'(x) = 6e^(9x)x^2(1+3x)`

`f(x) = 2x^3e^(9x) = h(x)g(x)`, where `h(x) = 2x^3` and `g(x) = e^(9x)`.

We will find the derivative with the product rule :

`f'(x) = h'(x)g(x) + h(x)g'(x) = (2x^3)'e^(9x)+2x^3(e^(9x))'`

`f'(x) = 2*3x^2*e^(9x) + 2x^3*9e^(9x) = 6x^2e^(9x) +18x^3e^(9x)`

`f'(x) = 6e^(9x)(x^2 + 3x^3) = 6e^(9x)x^2(1+3x)`