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The Reciprocal Rule

April 9th, 2009 |

Author: Merrin

The reciprocal rule plays a role in the derivation of the quotient rule as well as developing more complicated rules which obscure the limit definition of the derivative.

Example 1. Find the derivative of the reciprocal of x.

Solution 1. We can find the derivative by applying the definition of the derivative.

$\cfrac{d}{dx} \cfrac{1}{x} =\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{\cfrac{1}{x+\Delta x} -\cfrac{1}{x} }{\Delta x} =\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{-\Delta x}{x(x+\Delta x)\Delta x} =-\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{1}{x(x+\Delta x)} =-\cfrac{1}{x^{2} }$

Using this result and with our knowledge of the chain rule we can substitute a function u(x) for x and write.

$\cfrac{d}{dx} \cfrac{1}{u} =-\cfrac{1}{u^{2} } \cfrac{du}{dx}$

Another way to write this is

$\cfrac{d}{dx} \cfrac{1}{f(x)} =-\cfrac{1}{f^{2} (x)} \cfrac{d}{dx} f(x)$

Method 2: This method does not require the chain rule.

$f= 1/u$

$fu=1$

Now differentiate both sides of the equation

$f'u+fu' = 0$

$f' = -fu'/u$

But since f= 1/u we can write the derivative as

$f'(x)= -\cfrac{1}{u^2}\cfrac{du}{dx}$

Example 2. Find the derivative of the reciprocal of

$f(x)=x^{n}$

Solution 2. Apply the formula we got from the chain rule and the derivative of the reciprocal of x.

$\cfrac{d}{dx} \cfrac{1}{f(x)} =\cfrac{d}{dx} \cfrac{1}{x^{n} } =-\cfrac{1}{(x^{n} )^{2} } \cfrac{d}{dx} x^{n} =-n\cfrac{x^{n-1} }{x^{2n} } =-nx^{-n-1}$

We have the result that

$\cfrac{d}{dx} x^{-n} =-nx^{-n-1}$

This is just a continuation of the power rule for negative integer exponents. Later we will see that the power rule hold for all real exponents with logarithmic differentiation.

Example 3. Find the derivative of exp(x), given that the derivative of exp(x) is exp(x).

Solution 3. By the reciprocal rule we have that

$\cfrac{d}{dx} \cfrac{1}{e^{x} } =-\cfrac{1}{e^{2x} } \cfrac{d}{dx} e^{x} =-\cfrac{1}{e^{2x} } e^{x} =-\cfrac{1}{e^{x} } =-e^{-x}$