# Reciprocal rule for derivatives

## Reciprocal rule formula

The reciprocal rule is very similar to the quotient rule, except that it can only be used with quotients in which the numerator is a constant. Here is the formula:

Given a function

???h(x)=\frac{a}{f(x)}???

then its derivative is

???h'(x)=-a\frac{f'(x)}{\left[f(x)\right]^2}???

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Given that the numerator is a constant and the denominator is any function, the derivative will be the negative constant, multiplied by the derivative of the denominator divided by the square of the denominator.

## Reciprocal rule for the derivative of a fraction

Example

Use reciprocal rule to find the derivative.

???h(x)=\frac{1}{2x+1}+\frac{5}{3x-1}???

Applying the reciprocal rule gives

???h'(x)=-\frac{2}{(2x+1)^2}-\frac{15}{(3x-1)^2}???

The reciprocal rule is very similar to the quotient rule, except that it can only be used with quotients in which the numerator is a constant.

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