Leibniz's Integral Rule

Theorem
Let $\map f {x, t}$, $\map a t$, $\map b t$ be continuously differentiable real functions on some region $R$ of the $\tuple {x, t}$ plane.

Then for all $\tuple {x, t} \in R$:


 * $\ds \frac \d {\d t} \int_{\map a t}^{\map b t} \map f {x, t} \rd x = \map f {\map b t, t} \frac {\d b} {\d t} - \map f {\map a t, t} \frac {\d a} {\d t} + \int_{\map a t}^{\map b t} \frac \partial {\partial t} \map f {x, t} \rd x$

Proof
By the Mean Value Theorem for Integrals, there exist $\xi_a \in \closedint {\map a t} {\map a {t + h} }$ and $\xi_b \in \closedint {\map b t} {\map b {t + h} }$ such that:

Now, consider the last term:
 * $\ds \lim_{h \mathop \to 0} \int_{\map a t}^{\map b t} \frac {\map f {x, t + h} - \map f {x, t} } h \rd x$

By Limit of Function by Convergent Sequences, it suffices to find the following for an arbitrary sequence $\sequence {h_n}_{n \mathop \in \N}$ such that $\ds \lim_{n \mathop \to \infty} h_n = 0$:
 * $\ds \lim_{n \mathop \to 0} \int_{\map a t}^{\map b t} \frac {\map f {x, t + h_n} - \map f {x, t} } {h_n} \rd x$

Fix such as sequence, as well as a value for $t$.

For each $n \in \N$, define a function $\map {f_n} x$ as:
 * $\map {f_n} x = \frac {\map f {x, t + h_n} - \map f {x, t} } {h_n}$

when $\tuple {x, t}$ and $\tuple {x, t + h_n}$ are in $R$.

By the Extreme Value Theorem:
 * $M = \sup_{\tuple x \in R_t} \size {\frac \partial {\partial t} \map f {x, t} }$

is well-defined, where $R_t$ is the subset of $R$ where $t$ matches our chosen value.

Then, from the Mean Value Theorem, it follows that:
 * $\size {\map {f_n} x} \le M$

for all values of $n$ and $x$.

By Limit of Function by Convergent Sequences, as $n \to \infty$:
 * $\ds \map {f_n} x \to \lim_{h \mathop \to 0} \frac {\map f {x, t + h} - \map f {x, t} } h = \frac \partial {\partial t} \map f {x, t}$

by definition of Partial Derivative.

Therefore, by Lebesgue's Dominated Convergence Theorem:
 * $\ds \lim_{n \mathop \to 0} \int_{\map a t}^{\map b t} \frac {\map f {x, t + h_n} - \map f {x, t} } {h_n} \rd x = \int_{\map a t}^{\map b t} \frac \partial {\partial t} \map f {x, t} \rd x$

Combining with the result from before:
 * $\ds \frac \d {\d t} \int_{\map a t}^{\map b t} \map f {x, t} \rd x = \frac {\d b} {\d t} \map f {\map b t, t} - \frac {\d a} {\d t} \map f {\map a t, t} + \int_{\map a t}^{\map b t} \frac \partial {\partial t} \map f {x, t} \rd x$

Also known as
This is also referred to in some sources as Leibniz's Rule, but as this name is also used for a different result, it is necessary to distinguish between the two.

Other popular names for this technique include differentiation under the integral sign and Feynman's technique after physicist Richard Feynman.

Also see

 * Definite Integral of Partial Derivative, where $\map a t$ and $\map b t$ are constant:


 * $\ds \frac {\rd} {\rd t} \int_a^b \map f {x, t} \rd x = \int_a^b \frac {\partial} {\partial t} \map f {x, t} \rd x$