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Quotient rule

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In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist.

Contents

If the function one wishes to differentiate, f ( x ) , can be written as

f ( x ) = g ( x ) h ( x )

and h ( x ) ≠ 0 , then the rule states that the derivative of g ( x ) / h ( x ) is

f ′ ( x ) = g ′ ( x ) h ( x ) − g ( x ) h ′ ( x ) [ h ( x ) ] 2 .

Proof using implicit differentiation

Let f ( x ) = g ( x ) h ( x ) Then g ( x ) = f ( x ) h ( x )   Using product rule, g ′ ( x ) = f ′ ( x ) h ( x ) + f ( x ) h ′ ( x )   f ′ ( x ) = g ′ ( x ) − f ( x ) h ′ ( x ) h ( x ) = g ′ ( x ) ⋅ h ( x ) h ( x ) − g ( x ) h ( x ) ⋅ h ′ ( x ) h ( x ) f ′ ( x ) = g ′ ( x ) h ( x ) − g ( x ) h ′ ( x ) ( h ( x ) ) 2

Proof using chain rule

f ( x ) = g ( x ) h ( x )

We rewrite the fraction using a negative exponent.

f ( x ) = g ( x ) ⋅ h ( x ) − 1

Take the derivative of both sides, and apply the product rule to the right side.

f ′ ( x ) = g ′ ( x ) ⋅ h ( x ) − 1 + g ( x ) ⋅ ( h ( x ) − 1 ) ′

To evaluate the derivative in the second term, apply the chain rule, where the outer function is x − 1 , and the inner function is h ( x ) .

f ′ ( x ) = g ′ ( x ) ⋅ h ( x ) − 1 + g ( x ) ⋅ ( − 1 ) h ( x ) − 2 h ′ ( x )

Rewrite things in fraction form.

f ′ ( x ) = g ′ ( x ) h ( x ) − g ( x ) ⋅ h ′ ( x ) [ h ( x ) ] 2 f ′ ( x ) = g ′ ( x ) ⋅ h ( x ) − h ′ ( x ) ⋅ g ( x ) [ h ( x ) ] 2

Higher order formulas

It is much easier to derive higher order quotient rules using implicit differentiation. For example, two implicit differentiations of f h = g yields f ″ h + 2 f ′ h ′ + f h ″ = g ″ and solving for f ″ yields

f ″ = g ″ − 2 f ′ h ′ − f h ″ h .

Mnemonics

Many people remember the quotient rule by the rhyme "Low D-high, high D-low, cross the line and square below." "Low" refers to the denominator of the fraction, "high" refers to the numerator, and "D" means derivative.

References

Quotient rule Wikipedia
(Text) CC BY-SA


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