Question

How do you differentiate `xe^x`?

Answer 1

Use the product rule.

Explanation:

The Product rule (for derivatives) says that for differentiable functions, `f` and `g`, the derivative of the product is given by:
`d/dx(f(x)g(x)) = f'(x)g(x)+f(x)g'(x)`

In this question, we have

`f(x) = x`, so `f'(x) = 1`, and

`g(x) = e^x`, so `g'(x)=e^x`

`d/dx(f(x)g(x)) = (1)(e^x)+x(e^x)`

`= e^x+xe^x` `" "` which you may prefer to write as

`= e^x(1+x)` or as `e^x(x+1)` or in some other way.