Show through chain rule that $(u\cdot v)' = uv' + v'u$

Question

Let function be $f(x)=u\cdot v$ where $u$ and $v$ are in terms of $x$. Then how to make someone understand that $f'(x) = uv' + u'v $ only using chain rule?

My attempt: I don't even think it is possible but I may be wrong.

Answer 1

We have the multivariate chain rule:

$$\frac d{dx}f(u,v)=\frac{\partial f}{\partial u}\frac{du}{dx}+\frac{\partial f}{\partial v}\frac{dv}{dx}$$

and with $f(u,v)=u\cdot v$, we get

$$\frac d{dx}u\cdot v=vu'+uv'$$

since

$$\frac\partial{\partial u}u\cdot v=v$$

$$\frac\partial{\partial v}u\cdot v=u$$