In calculus, the reciprocal rule gives the derivative of the reciprocal of a function f in terms of the derivative of f. The reciprocal rule can be used to show that the power rule holds for negative exponents if it has already been established for positive exponents.
Also, one can readily deduce the quotient rule from the reciprocal rule and the product rule.
The reciprocal rule states that if f is differentiable at a point x and f(x) ≠ 0 then g(x) = 1/f(x) is also differentiable at x and
This proof relies on the premise that
is then also necessarily continuous there.
Applying the definition of the derivative of
The limit of this product exists and is equal to the product of the existing limits of its factors:
the first limit equals
the second limit equals
an application of the product rule says that
and this may be algebraically rearranged to say
However, this fails to prove that 1/f is differentiable at x; it is valid only when differentiability of 1/f at x is already established.
In that way, it is a weaker result than the reciprocal rule proved above.
However, in the context of differential algebra, in which there is nothing that is not differentiable and in which derivatives are not defined by limits, it is in this way that the reciprocal rule and the more general quotient rule are established.
Often the power rule, stating that
, is proved by methods that are valid only when n is a nonnegative integer.
This can be extended to negative integers n by letting
, where m is a positive integer.
by the reciprocal rule
by the power rule applied to the positive integer
by substituting back
{\displaystyle {\begin{aligned}{\frac {d}{dx}}x^{n}&={\frac {d}{dx}}\,\left({\frac {1}{x^{m}}}\right)\\&=-{\frac {{\frac {d}{dx}}x^{m}}{(x^{m})^{2}}},{\text{ by the reciprocal rule}}\\&=-{\frac {mx^{m-1}}{x^{2m}}},{\text{ by the power rule applied to the positive integer }}m,\\&=-mx^{-m-1}=nx^{n-1},{\text{ by substituting back }}n=-m.\end{aligned}}}
The reciprocal rule is a special case of the quotient rule, which states that if f and g are differentiable at x and g(x) ≠ 0 then
The quotient rule can be proved by writing
and then first applying the product rule, and then applying the reciprocal rule to the second factor.
By using the reciprocal rule one can find the derivative of the secant and cosecant functions.
For the secant function:
= sec x tan x .
The cosecant is treated similarly:
= − csc x cot x .