A proof of the quotient rule

6 February 2018 in MAE 113: Calculus for the Liberal Arts | No comments

The quotient rule proof can be quite tedious if we are only allowed to use the definition of the derivative.

The Quotient Rule: \left( \dfrac{f(x)}{g(x)}\right)' = \dfrac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}

Let h(x) = \displaystyle\frac{f(x)}{g(x)}. Then, using the definition of the derivative we have

    \begin{align*} &h'(x) \\ &= \lim_{\Delta x \to 0} \frac{h(x + \Delta x) - h(x)}{\Delta x}\\ &= \lim_{\Delta x \to 0} \frac{\frac{f(x + \Delta x)}{g(x + \Delta x)} - \frac{f(x)}{g(x)}}{\Delta x}\\ &= \lim_{\Delta x \to 0} \frac{1}{\Delta x}\left(\frac{f(x + \Delta x)}{g(x + \Delta x)} - \frac{f(x)}{g(x)}\right)\\ &= \lim_{\Delta x \to 0} \frac{1}{\Delta x}\left(\frac{f(x + \Delta x)}{g(x + \Delta x)}\cdot \frac{g(x)}{g(x)} - \frac{g(x + \Delta x)}{g(x + \Delta x)} \cdot \frac{f(x)}{g(x)}\right)\\ &= \lim_{\Delta x \to 0} \frac{1}{\Delta x}\left(\frac{f(x + \Delta x)g(x) - g(x + \Delta x) f(x)}{g(x + \Delta x) g(x)}\right)\\ &= \lim_{\Delta x \to 0} \frac{1}{g(x + \Delta x) g(x)}\left(\frac{f(x + \Delta x)g(x) - g(x + \Delta x) f(x)}{\Delta x}\right)\\ &= \lim_{\Delta x \to 0} \frac{1}{g(x + \Delta x) g(x)}\\ & \left(\frac{f(x + \Delta x)g(x) - f(x)g(x) + f(x)g(x) - g(x + \Delta x) f(x)}{\Delta x}\right)\\ &= \lim_{\Delta x \to 0} \frac{1}{g(x + \Delta x) g(x)} \left(g(x) \frac{f(x + \Delta x) - f(x)}{\Delta x}\right.\\ &\qquad \left. - f(x) \frac{g(x + \Delta x) + g(x)}{\Delta x}\right)\\ &= \lim_{\Delta x \to 0} \frac{1}{g(x + \Delta x) g(x)}\left(\lim_{\Delta x \to 0} g(x) \lim_{\Delta x \to 0}\frac{f(x + \Delta x) - f(x)}{\Delta x}\right.\\ &\qquad\qquad \left. - \lim_{\Delta x \to 0} f(x) \lim_{\Delta x \to 0}\frac{g(x + \Delta x) + g(x)}{\Delta x}\right)\\ &= \lim_{\Delta x \to 0} \frac{1}{g(x + \Delta x) g(x)}\big(g(x) f'(x) - f(x) g'(x)\big)\\ &= \frac{1}{g(x) g(x)}\big(f'(x) g(x) - f(x) g'(x)\big)\\ &= \frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2} \end{align*}

That was quite long. But, we have proved the quotient rule.

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