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Calculus β

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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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}

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