What is the derivative of $xf(x)$?

Question

If we have a function $f(x)$ what would the value of the following expression be? $$\frac{d}{dx}[xf(x)]$$

Answer 1

Hint:

What would you do if the expression was $$\frac{d}{dx}\left [g(x)f(x)\right ]$$?

Answer 2

If x is a variable, x(f(x)) is the product x f(x) and the standard product rule applies.

If x is a function, you have to define what x(f(x)) means and then the chain rule applies: (x(f(x)))' = f'(x)x'(f(x)) so you also have to define what x'(f(x)) means.

The notation x(f(x)) is certainly non-standard.