In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let , where both f and g are differentiable and The quotient rule states that the derivative of h(x) is
It is provable in many ways by using other derivative rules.
Given , let , then using the quotient rule:
The quotient rule can be used to find the derivative of as follows:
The reciprocal rule is a special case of the quotient rule in which the numerator . Applying the quotient rule gives
Note that utilizing the chain rule yields the same result.
Let Applying the definition of the derivative and properties of limits gives the following proof, with the term added and subtracted to allow splitting and factoring in subsequent steps without affecting the value:
Let so The product rule then gives Solving for and substituting back for gives:
Let Then the product rule gives
Substituting the result into the expression gives
Let Taking the absolute value and natural logarithm of both sides of the equation gives
Implicit differentiation can be used to compute the nth derivative of a quotient (partially in terms of its first n − 1 derivatives). For example, differentiating twice (resulting in ) and then solving for yields