Proof of the Quotient Rule
to accompany
Calculus Applied to the Real World

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The Quotient Rule

If the functions $f$ and $g$ are differentiable at $x,$ with $g(x) ≠ 0,$ then the quotient $\frac{f}{g}$ is differentiable at $x,$ and

    $\frac{d}{dx}\ \Bigl\[ \frac{f}{g} \Bigr\]\ (x) = \frac{f^{'} (x) g(x) - f(x) g^{'}(x)}{[g(x)]^2.}$

Proof By the definition of the derivative,

If we recognize the difference quotients for $f$ and $g$ in this last expression, we see that taking the limit as $h \to 0$ replaces them by the derivatives $f^{'}(x)$ and $g^{'}(x).$ Further, since $g$ is differentiable, it is also continuous, and so $g(x+h) \to g(x)$ as $h \to 0.$ Putting this all together gives

which is the quotient rule.