Question

What is the derivative of `y=3x^2e^(5x)` ?

Answer 1

This is a product of the function `3x^2` and the function `e^(5x)`, which is itself the composite of the functions given by `e^x` and by `5x`.

Thus we will need the product rule to the effect that:

`(3x^2 e^(5x))'=(3x^2)'e^(5x)+3x^2(e^(5x))'`

As well as the fact that:

`(3x^2)'=6x`

Using these basic differentiation rules, and the Chain Rule combined with `(e^x)'=e^x`, we can calculate that:

`(e^(5x))'=e^(5x)\times (5x)'=5e^(5x)`.

As a result:

`(3x^2 e^(5x))'=6xe^(5x)+15x^2e^(5x)`.