Question

How do you differentiate `y=xe^x`?

Answer 1

`frac{"d"}{"d"x}(xe^x) = (1 + x) e^x`

Use the product rule.

Explanation:

The product rule:

If `u` and `v` are differentiable functions of `x`,

and `f = u * v`,

then `f' = u' * v + u * v'`,

where the apostrophe denotes the derivative with respect to `x`.

In the above question, we can see that `x e^x` is a product of `x` and `e^x`, both which are elementary functions.

Thus

`frac{"d"}{"d"x}(xe^x) = frac{"d"}{"d"x}(x) e^x + x frac{"d"}{"d"x}(e^x)`

`= (1) e^x + x (e^x)`

`= (1 + x) e^x`