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.
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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 $g(x_0) \ne 0$. 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