Quotient rule - Knowlpedia

If $f,g$ are differentiable at $x$ and $g(x)\ne0$ , then $g$ stays nonzero near $x$ and

\left(\frac fg\right)'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}.

This follows by applying the product rule to $f\cdot g^{-1}$ and the chain rule to the reciprocal. The same formula holds for each partial derivative.

Vanishing denominators

The formula does not define a quotient at a zero of $g$ . A separate extension argument may remove a particular singularity, for example $r^{-1}\partial_r a(r^2)=2a'(r^2)$ for smooth $a$ . Such cancellation must be established before claiming regularity on the axis.