Quotient rule for differentiation

This article is about a differentiation rule, i.e., a rule for differentiating a function expressed in terms of other functions whose derivatives are known.
View other differentiation rules

Statement

Suppose $f$ and $g$ are functions defined at and around a point $x_{0}$ and they are both differentiable at $x_{0}$ (i.e., the derivatives $f'(x_{0})$ and $g'(x_{0})$ are defined) and $g(x_{0})\neq 0$. Then, the quotient $f/g$ is differentiable at $x_{0}$, and its derivative is given as follows:

${\frac {d}{dx}}\left({\frac {f(x)}{g(x)}}\right)|_{x=x_{0}}={\frac {g(x_{0})f'(x_{0})-f(x_{0})g'(x_{0})}{(g(x_{0}))^{2}}}$

If we consider the general expressions rather than the evaluation at a particular point $x_{0}$, we can rewrite the above as:

${\frac {d}{dx}}\left({\frac {f(x)}{g(x)}}\right)={\frac {g(x)f'(x)-f(x)g'(x)}{(g(x))^{2}}}$

Related rules

• Product rule for differentiation
• Second derivative rule for parametric descriptions (uses the quotient rule in its proof)
• Quotient rule for second derivative