Let
- $f$ be a continuous and differentiable function
- $\vec{u}=<x, f(x)>$
- $\vec{v}=<x(t),f(t)>$
- $x(t)=t$
Then, $$\vec{u}=\vec{v}$$ $$\frac{d\vec{u}}{dx}=\frac{d\vec{v}}{dt}$$ $$\frac{df}{dx}=\frac{\frac{df}{dt}}{\frac{dx}{dt}}$$ $$\frac{df}{dt}=\frac{df}{dx}\frac{dx}{dt}$$
I've seen lots of other proofs that important subtleties need to be considered whereas here they seem to vanish simply because $\frac{dx}{dt}=1$.
Thanks for any feedback :)