In
calculus, the quotient rule is a method of finding the
derivative of a
function that is the ratio of two differentiable functions.[1][2][3] Let , where both f and g are differentiable and The quotient rule states that the derivative of h(x) is
Proof from derivative definition and limit properties
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:The limit evaluation is justified by the differentiability of , implying continuity, which can be expressed as .
Taking the absolute value of the functions is necessary for the
logarithmic differentiation of functions that may have negative values, as logarithms are only
real-valued for positive arguments. This works because , which justifies taking the absolute value of the functions for logarithmic differentiation.
Higher order derivatives
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
See also
Chain rule – For derivatives of composed functions
In
calculus, the quotient rule is a method of finding the
derivative of a
function that is the ratio of two differentiable functions.[1][2][3] Let , where both f and g are differentiable and The quotient rule states that the derivative of h(x) is
Proof from derivative definition and limit properties
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:The limit evaluation is justified by the differentiability of , implying continuity, which can be expressed as .
Taking the absolute value of the functions is necessary for the
logarithmic differentiation of functions that may have negative values, as logarithms are only
real-valued for positive arguments. This works because , which justifies taking the absolute value of the functions for logarithmic differentiation.
Higher order derivatives
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
See also
Chain rule – For derivatives of composed functions