Partial Derivative / Multivariable Chain Rule Notation

Question

Example 1.

w = $x^2 +y^2 + z^2$

x = $uv$

y = $ucos(v)$

z = $ usin(v)$

Using chain rule, find $\frac{∂w}{∂u}$

I understand partial derivatives and how to take chain rule for multivariable functions but why does $\frac{∂w}{∂u}$imply the same meaning as $\frac{∂w∘f}{∂u}$ where$ f(u,v) = (x,y,z)$

Answer 1

Because when you write “w” (and implicitly are thinking of x, y and z as functions of u and v) you are already thinking of $w \circ f$. No difference between the two.