Question

How do you differentiate `f(x)=e^(-x)sinx` using the product rule?

Answer 1

`f'(x) = e^(-x) (cos x - sin x)`

Explanation:

The product rule is as follows:
if your function is a product of two functions,

`f(x) = g(x) * h(x)`

then the derivative is

`f'(x) = g'(x) * h(x) + g(x) * h'(x)`

In your case, `g(x) = e^(-x)` and `h(x) = sin x`.

Let's find the derivatives `g'(x)` and `h'(x)`:

`g(x) = e^(-x) color(white)(xxx) => color(white)(x) g'(x) = -e^(-x)`

`h(x) = sin x color(white)(xxi) => color(white)(x) h'(x) = cos x`

So, in total, your derivative is:

`f'(x) = g'(x) * h(x) + g(x) * h'(x)`

`color(white)(xxxx) = -e^(-x) sin x + e^(-x) cos x`

`color(white)(xxxx) = color(white)(x) e^(-x) (cos x - sin x)`