differentiation under the integral sign

The technique of differentiationMathworldPlanetmath under the integral sign concerns the interchange of the operationMathworldPlanetmath of differentiation with respect to a parameter with the operation of integration over some other variable:


Intuitively, the rule ought to work because differentiation commutes with finite summation, and one may conjecture that it can also commute with infiniteMathworldPlanetmathPlanetmath summation (in the form of the integralDlmfPlanetmath), at least in some cases.

The theorems below give some sufficient conditions, in increasing generality and sophistication, for which the swap of differentiation and integration is legal.

Formal statements

Theorem 1 (Elementary Calculus version).

Let f:[a,b]×YR be a functionMathworldPlanetmath, with [a,b] being a closed intervalDlmfMathworldPlanetmath, and Y being a compact subset11Assumed to be Jordan-measurable if the Riemann integral is to be used. of Rn. Suppose that both f(x,y) and f(x,y)/x are continuousMathworldPlanetmathPlanetmath in the variables x and y jointly. Then Yf(x.y)dy exists as a continuously differentiable function of x on [a,b], with derivativePlanetmathPlanetmath


Theorem 1 is the formulation of integration under the integral sign that usually appears in elementary Calculus texts. Unfortunately, its restrictionPlanetmathPlanetmath that Y must be compact can be quite severe for applications: e.g. integrals over (-,+) are not included. Theorem 2 below addresses this problem and others:

Theorem 2 (Measure theory version).

Let X be an open subset of R, and Ω be a measure spaceMathworldPlanetmath. Suppose f:X×ΩR satisfies the following conditions:

  1. 1.

    f(x,ω) is a Lebesgue-integrable function of ω for each xX.

  2. 2.

    For almost all ωΩ, the derivative f(x,ω)/x exists for all xX.

  3. 3.

    There is an integrable function Θ:Ω such that |f(x,ω)/x|Θ(ω) for all xX.

Then for all xX,


Theorem 2 suffices for many applications, but using the Fundamental Theorem of Calculus for Lebesgue integration, we can weaken the hypotheses for differentiating under the integral sign even further:

Theorem 3.

Let X be an open subset of R, and Ω be a measure space. Suppose that a function f:X×ΩR satisfies the following conditions:

  1. 1.

    f(x,ω) is a measurable functionMathworldPlanetmath of x and ω jointly, and is integrable over ω, for almost all xX held fixed.

  2. 2.

    For almost all ωΩ, f(x,ω) is an absolutely continuous function of x. (This guarantees that f(x,ω)/x exists almost everywhere.)

  3. 3.

    f/x is “locally integrable” — that is, for all compact intervals [a,b] contained in X:


Then Ωf(x,ω)𝑑ω is an absolutely continuous function of x, and for almost every xX, its derivative exists and is given by


If the Kurzweil-Henstock integral — which has a stronger Fundamental Theorem of CalculusMathworldPlanetmath (http://planetmath.org/FundamentalTheoremOfCalculusForKurzweilHenstockIntegral) — is used in place of the Lebesgue integral, Theorem 3 can be generalized to a formulation that provides also the necessary conditions for differentiation under the integral sign. See [Talvila] for the full details.

Yet this is not the end of the story. There are some applications in which the integrand is too “irregular”, or the integral of the differentiated integrand becomes divergent, and neither Theorem 2 or Theorem 3 would apply. However, if we use generalized functions (all of which can be differentiated at will), then we can extend the technique of differentiation under the integral sign further, and make sense of any “irregular” integrals that may result:

Theorem 4 (Distribution theory version).

Let X be an open set in Rm, and Ω be a measure space. Given f(x,ω), for each ωΩ, a generalized function of xX (in the sense of Schwartz’s theory of distributionsDlmfPlanetmath), define:


Assume the above integral is well-defined and gives a distribution. Then


where /xi refers to the generalized derivative of generalized functions on both sides of the equation.

For an absolutely continuous function, the generalized derivative coincides with the ordinary derivative, so Theorem 4 indeed generalizes Theorem 3. On the other hand, there are cases where the integrand is not absolutely continuousMathworldPlanetmath — and so has a generalized derivative different from the ordinary derivative — yet its integral has a classical derivative that is represented by the final equation of Theorem 4. For instance, the integrand may involve a step functionPlanetmathPlanetmath, and its derivative would thus involve a Dirac delta distribution, that when integrated, yields an ordinary locally-integrable function (of the parameter x).

Theorem 4 is not so well-publicized, but appears, for example, in [Jones], and hinted at in a comment in [Schwartz].

Other variations

There are other frequently-used variations of the theorems above.

Moving domains of integration. Not only can the integrand vary with the parameter, we can consider domains of integration, subsets of n, that vary with the parameter.

In the one-dimensional case, for continuously differentiable functions α:[a,b], β:[a,b], and f:[a,b]×, we have:


This result can be extrapolated from Theorem 1, with the help of the Fundamental Theorem of Calculus and the multi-variate chain ruleMathworldPlanetmath (http://planetmath.org/ChainRuleSeveralVariables).

GeneralizationsPlanetmathPlanetmath to varying smooth surfaces or volumes — or, more generally, k-dimensional differentiable manifolds in n — can be obtained by using integrals of differential formsMathworldPlanetmath on chains (http://planetmath.org/NChain), and Stokes’ Theorem. Details can be found in [Flanders].

Different types of integrals. The differentiation can also be taken under integrals other than of the standard Riemann type, such as the line integralsPlanetmathPlanetmath and surface integrals of vector calculus, or complex contour integrals. (Actually, these kinds of integrals can be re-formulated as Lebesgue integrals, so Theorem 2 applies to them.)

Complex variables. Other applications require differentiating holomorphic functionsMathworldPlanetmath with respect to a complex variable, and Theorem 2 generalizes directly to this situation, without requiring differentiation with respect to real variables as an intermediary.


  • Flanders Harley Flanders. “Differentiation under the Integral Sign”. American Mathematical Monthly, vol. 80 (June-July 1973), p. 615-627.
  • Folland Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications, second ed. Wiley-Interscience, 1999.
  • Jones D. S. Jones. The Theory of Generalized Functions, second ed. Cambridge University Press, 1982.
  • Munkres James R. Munkres. Analysis on Manifolds. Westview Press, 1991.
  • Schwartz Laurent Schwartz. Théorie des Distributions, vol. I. Hermann, 1957.
  • Talvila Erik Talvila. “http://www.math.ualberta.ca/ etalvila/papers/difffinal.pdfNecessary and Sufficient Conditions for Differentiating Under the Integral Sign”. American Mathematical Monthly, vol. 108 (June-July 2001), p. 544-548.

The author of this entry has also written an exposition, “http://gold-saucer.afraid.org/math/diff-int/diff-int.pdfDifferentiation under the Integral Sign using Weak Derivatives”, containing a proof of Theorem 4 along with detailed computational examples.

Title differentiation under the integral sign
Canonical name DifferentiationUnderTheIntegralSign
Date of creation 2013-03-22 16:26:37
Last modified on 2013-03-22 16:26:37
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 16
Author stevecheng (10074)
Entry type Topic
Classification msc 46F10
Classification msc 28A25
Classification msc 26B15
Classification msc 26A24
Synonym Leibniz’s rule
Related topic DerivativeOfRiemannIntegral
Related topic IntegrationUnderIntegralSign
Related topic HolomorphicFunctionAssociatedWithContinuousFunction