How I do I prove the Quotient Rule for derivatives?

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Answer 1 · 2016-04-30T20:14:51

We can use the product rule:

$d/dx[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$

In order to prove the quotient rule, which states that

$d/dx[f(x)/g(x)]=(f'(x)g(x)-f(x)g'(x))/(g(x))^2$

However, we can apply this to the product rule by writing $f(x)/g(x)$ as $f(x)(g(x))^-1$ .

Use the product rule on this:

$d/dx[f(x)(g(x))^-1]=f'(x)(g(x))^-1+f(x)d/dx[(g(x))^-1]$

Differentiating $d/dx[(g(x))^-1]$ requires the chain rule:

$d/dx[(g(x))^-1]=-g'(x)(g(x))^-2$

Hence we obtain

$d/dx[f(x)(g(x))^-1]=f'(x)(g(x))^-1-f(x)g'(x)(g(x))^-2$

Rewriting with fractions, this becomes

$d/dx[f(x)/g(x)]=(f'(x))/g(x)-(f(x)g'(x))/(g(x))^2$

Rewriting with a common denominator, this becomes

$d/dx[f(x)/g(x)]=(f'(x)g(x)-f(x)g'(x))/(g(x))^2$