If $f = f(x,y)$ and $C$ is constant, then is this true: $\frac{\partial f}{\partial (Cx)}=\frac{1}{C}\frac{\partial f}{\partial x}$?

Question

Suppose $f=f(x,y)$ is any function of 2 variables and $C=\mbox{const}$. Can one do

$$\frac{\partial f}{\partial (Cx)}=\frac{1}{C}\frac{\partial f}{\partial x}$$

Is this always correct? Or do we need the derivative $f_x'$ to exist, or be continuous before we can write the above?

Answer 1

If by "constant" you mean $C$ depends on neither $x$ nor $y$, then for $C\ne 0$ you only need $\partial f/\partial x$ to exist. Necessity is trivial; sufficiency is an exercise. (Hint: use $\lim_{x\to a}kg(x)=k\lim_{x\to a}g(x)$.)