Chapter 3.4, Problem 86E

Proof of the Quotient Rule Let F = f/g be the quotient of two functions that are differentiable at x.

1. a. Use the definition of F′ to show that

$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\underset{h\to 0}{\lim }\frac{f(x+h)g(x)-f(x)g(x+h)}{hg(x+h)g(x)}.$

1. b. Now add −f(x)g(x) + f(x)g(x) (which equals 0) to the numerator in the preceding limit to obtain

$\underset{h\to 0}{\lim }\frac{f(x+h)g(x)-f(x)g(x)+f(x)g(x)-f(x)g(x+h)}{hg(x+h)g(x)}$

Use this limit to obtain the Quotient Rule.

1. c. Explain why F′ = (f/g)′ exists, whenever g(x) ≠ 0.