How do you prove the quotient rule?

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Answer 1 · 2014-10-02T09:47:24

By the definition of the derivative,

$[{f(x)}/{g(x)}]'=lim_{h to 0}{f(x+h)/g(x+h)-f(x)/g(x)}/{h}$

by taking the common denominator,

$=lim_{h to 0}{{f(x+h)g(x)-f(x)g(x+h)}/{g(x+h)g(x)}}/h$

by switching the order of divisions,

$=lim_{h to 0}{{f(x+h)g(x)-f(x)g(x+h)}/h}/{g(x+h)g(x)}$

by subtracting and adding $f(x)g(x)$ in the numerator,

$=lim_{h to 0}{{f(x+h)g(x)-f(x)g(x)-f(x)g(x+h)+f(x)g(x)}/h}/{g(x+h)g(x)}$

by factoring $g(x)$ out of the first two terms and $-f(x)$ out of the last two terms,

$=lim_{h to 0}{{f(x+h)-f(x)}/h g(x)-f(x){g(x+h)-g(x)}/h}/{g(x+h)g(x)}$

by the definitions of $f'(x)$ and $g'(x)$ ,

$={f'(x)g(x)-f(x)g'(x)}/{[g(x)]^2}$

I hope that this was helpful.