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The Quotient Rule

April 1st, 2009 |

Author: Merrin

Often in calculus, we are asked to find the derivative of one function divided by another. The differentiation rule for quotients is called the quotient rule. From our previous discussion on the product rule, you can see how the quotient rule is just the product rule in disguise, but the chain rule is required to differentiate the individual functions which may include a reciprocal. If you learn to use the quotient rule it will just speed up the last simplification step of your algebra compared to using the product rule in every case.

Theorem 1. The quotient rule, first form

$\cfrac{d}{dx} \left(\cfrac{f(x)}{g(x)} \right)=\cfrac{f'(x)g(x)-f(x)g'(x)}{g^{2}(x)}$

Theorem 1. The quotient rule, second form (less pen strokes)

$\cfrac{d}{dx} \left( \cfrac{u}{v} \right) = \cfrac{u'v-uv'}{v^2}$

Proof 1. The quotient rule can be derived using the product rule.

$z = \cfrac{u}{v}$

$zv = u$

Now we differentiate the product and try to solve for the derivative of z.

$\cfrac{dz}{dx}v+z\cfrac{dv}{dx}=\cfrac{du}{dx}$

$\cfrac{dz}{dx}v+\cfrac{u}{v}\cfrac{dv}{dx}=\cfrac{du}{dx}$

$z'= \cfrac{u'}{v}-\cfrac{u}{v^2}v'$

$z'=\cfrac{u'v-uv'}{v^2}$

Proof 2. The quotient rule can also be viewed as a use of the product rule where one function is a reciprocal.

$\cfrac{d}{dx} (u/v) = (u)'(v^{-1})+(u)(v^{-1})'$

$\cfrac{d}{dx} (u/v) = u'/v -uv'/v^2$

$\cfrac{d}{dx} (u/v) = \cfrac{u'v-uv'}{v^2}$

Example 1. Find the derivative of

$f(x)=\cfrac{\ln x}{x}$

Solution 1. If the individual derivatives of the numerator and denominator are known then simply plug them into the formula for the quotient rule. If it takes more than several lines work then you are doing something wrong.

$\cfrac{d}{dx} \left( \cfrac{\ln x}{x} \right) = \cfrac{(\ln x)'(x)-(\ln x)(x)'}{x^2}$

$\cfrac{d}{dx} \left( \cfrac{\ln x}{x} \right) = \cfrac{(x^{-1})(x)-(\ln x)(1)}{x^2}$

$\cfrac{d}{dx} \left( \cfrac{\ln x}{x} \right) = \cfrac{1-\ln x}{x^2}$

Example 2. Find the derivative of

$f(x)=\cfrac{x^{n}+a^{n}}{x^{n}+b^{n}}$

Solution 2. The derivatives of the individual functions in the numerator and denominator are known because they are just polynomials, so just apply the quotient rule formula.

$f'(x) = \cfrac{d}{dx} \cfrac{x^{n} +a^{n} }{x^{n} +b^{n} }$