In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, the reciprocal rule gives the derivative of the
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of a function ''f'' in terms of the derivative of ''f''. The reciprocal rule can be used to show that the
power rule
In calculus, the power rule is used to differentiate functions of the form f(x) = x^r, whenever r is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated usin ...
holds for negative exponents if it has already been established for positive exponents. Also, one can readily deduce the
quotient rule
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h(x)=f(x)/g(x), where both and are differentiable and g(x)\neq 0. The quotient rule states that the deriva ...
from the reciprocal rule and the
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
.
The reciprocal rule states that if ''f'' is
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
at a point ''x'' and ''f''(''x'') ≠ 0 then g(''x'') = 1/''f''(''x'') is also differentiable at ''x'' and
Proof
This proof relies on the premise that
is differentiable at
and on the theorem that
is then also necessarily
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
there. Applying the definition of the derivative of
at
with
gives
The limit of this product exists and is equal to the product of the existing limits of its factors:
Because of the differentiability of
at
the first limit equals
and because of
and the continuity of
at
the second limit thus yielding
A weak reciprocal rule that follows algebraically from the product rule
It may be argued that since
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 mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A natur ...
, 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.
Application to generalization of the power rule
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.
Application to a proof of the quotient rule
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.
Application to differentiation of trigonometric functions
By using the reciprocal rule one can find the derivative of the secant and cosecant functions.
For the secant function:
The cosecant is treated similarly:
See also
*
*
*
*
*
*
*
*
*
*
*
*
References
{{Calculus topics
Articles containing proofs
Differentiation rules
Theorems in analysis
Theorems in calculus