Is the generalized chain rule for partial derivatives with multiple independent variables only for convenience, or are there problems that require its use?
The questions I have seen so far are at worst like:
Let $w(x,y,z)=\sqrt{x^2+y^2+z^2}$ where $x=4re^t$, $y=-3te^r$, $z=e^{rt}$.
Find $\frac{\partial w}{\partial r}$ and $\frac{\partial w}{\partial t}$.
It doesn't seem like using the chain rule here saves much work (if any?), does it? Are there other instances that actually require the chain rule?
My favourite example is $F(t)=\int_a^uf(v,x)dx$ where $u$ and $v$ are functions of $t$.For example, $$F(t)=\int_{-1}^{3t+1}e^{tx^2}dx$$ Find (a)$F^{\prime}(t)$ (b)$F^{\prime}(0)$. The only way I know how do such a question is by Leibniz's Rule, which is just the chain rule for 2 variables, combined with differentiaion through an integral for differentiation with respect to $v$ and the fundamental theorem of calculus for differentiation with respect to $u$. I agree with you completely that the examples given in many texts are silly.