When I have some product of two functions, such as: $$g(x)*f(x),$$
I can take the derivative with respect to $f(x)$, which by the chain rule, yields
$$\frac{\partial [g(x)*f(x)]}{\partial f(x)} = \frac{\partial g(x)}{\partial x} \cdot \frac{1}{f'(x)} = \frac{g'(x)}{f'(x)}.$$
If $f(x)$ were treated like a fixed variable $u$ instead of a function, we can express
$$g(x)*u \rightarrow \frac{\partial [g(x)*u]}{\partial u} = g(x)$$
What is the difference between the two? Would it ever be appropriate to use the second method and treat it as a variable instead of a function? What would the interpretation of the second method be?