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,
and
Proof using implicit differentiation
LetProof using chain rule
We rewrite the fraction using a negative exponent.
Take the derivative of both sides, and apply the product rule to the right side.
To evaluate the derivative in the second term, apply the chain rule, where the outer function is
Rewrite things in fraction form.
Higher order formulas
It is much easier to derive higher order quotient rules using implicit differentiation. For example, two implicit differentiations of
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.