##  [Differentiation Under the Integral Sign](/differentiation-under-integral-sign-0) 

 Definition

The procedure of interchanging differentiation with respect to a parameter and integration over a measure space in a parameter-dependent integral, justified under conditions such as uniform convergence, dominated convergence, or absolute integrability of the derivative.

 

 

 

 

 

 





## Principle

Principle

The rule is that if the integrand f(x,t) and its partial derivative ∂f/∂t satisfy integrability and uniformity or domination hypotheses on the parameter region, then d/dt ∫ f(x,t) dx = ∫ ∂f/∂t (x,t) dx; domination and uniform control permit the exchange of limit operations.

 

 

 

 

 





## Demonstration

Demonstration

Consider I(t)=∫_a^b e^{-t x^2} dx. The partial derivative ∂/∂t e^{-t x^2} = −x^2 e^{-t x^2} is dominated by x^2 e^{-t0 x^2} on a compact t-interval, so differentiate under the integral to compute I'(t)=∫_a^b −x^2 e^{-t x^2} dx.

 

 

 

 

## Misapplication

Misapplication

Interchanging differentiation and integration when ∂f/∂t fails to be integrable or lacks a dominating integrable bound; for instance, parameterized integrands with increasingly concentrated spikes can break the rule and give incorrect derivatives.

 

 

 

 

 





## Consequence

Consequence

When valid, the technique simplifies evaluation of parameter derivatives, gives access to differential identities for integrals, and underlies many analytic calculations in applied and pure analysis.

 

 

 

 

## Reversal

Reversal

The contrast is integrating the derivative without justification: computing ∫ ∂f/∂t when the interchange law fails may produce a value that does not equal the derivative of the integral; conversely, pointwise differentiation of integrals without control is invalid.

 

 

 

 

 





## Boundary

Boundary

Requires a precise measure-theoretic setting: integrable domain, parameter topology, and conditions like dominated convergence theorem, uniform convergence on compact parameter sets, or absolute integrability of ∂f/∂t. Does not apply to arbitrary pointwise derivatives or non-measurable integrands.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between the formal Leibniz rule (often taught heuristically) and measure-theoretic justification: formal symbol-manipulation can mislead if domination or uniformity hypotheses are omitted.

 

 

 

 

 





## Synthesis

Synthesis

Differentiation under the integral sign is the controlled interchange of differentiation and integration based on domination or uniform convergence hypotheses, enabling one to pass the derivative through the integral and thereby compute parameter derivatives reliably.