Question

What is the derivative of `f(x)=x sin (x)`?

Answer 1

`f'(x) = sinx+xcosx`

Explanation:

The product rule tells us that for `f(x) = uv` `" "` for functions `u` and `v`,

we get

`f'(x) = u'v+uv'`

For `f(x) = xsinx` we have `u = x` and `v = sinx`.

Apply the product rule:

`f'(x) = overbrace((1))^(u') overbrace((sinx))^v+overbrace((x))^u overbrace((cosx))^(v')`

`= sinx+xcosx`

I did notice that this was asked under "Differentiationg sin(x) from First Principles". If you really want to see this done using the limit definition (First Principles), ask and I'll type it up".