Leibniz Integral Rule

Math Calculus

Theorem

Let $f(x,t)$ be such that both $f(x,t)$ and its partial derivative $f_x(x,t)$ be continuous in $t$ and $x$ in a region of the $xt$-plane, such that $a(x)\le t\le b(x)$, $x_0\le x\le x_1$. Also let $a(x)$ and $b(x)$ be continuous and have continuous derivatives for $x_0\le x\le x_1$. Then, for $x_0\le x\le x_1$:

$$\frac{d}{dx}({\int }_{a(x)}^{b(x)}f(x,t)dt)=f(x,b(x))\cdot \frac{d}{dx}b(x)-f(x,a(x))\cdot \frac{d}{dx}a(x)+{\int }_{a(x)}^{b(x)}\frac{\partial }{\partial x}f(x,t)dt$$

Notably, this also means:

$$\frac{d}{dx}({\int }_{c_1}^{c_2}f(x)dx)={\int }_{c_1}^{c_2}\frac{d}{dx}f(x)dx$$

Proof

Let $\phi (x)={\int }_a^{b}f(x,t)dt$ where $a$ and $b$ are functions of $x$i. Define $\Delta a=a(x+\Delta x)-a(x)$ and $\Delta b=b(x+\Delta x)-b(x)$. Then,

$$\begin{gathered} \Delta \phi =\phi (x+\Delta x)-\phi (x) \\ ={\int }_{a+\Delta a}^{b+\Delta b}f(x+\Delta x,t)dt-{\int }_a^{b}f(x,t)dt \end{gathered}$$

Now expand the first integral by integrating over 3 separate ranges:

$$\begin{gathered} {\int }_{a+\Delta a}^{a}f(x+\Delta x,t)dt+{\int }_a^{b}f(x+\Delta x,t)dt+{\int }_b^{b+\Delta b}f(x+\Delta x,t)dt-{\int }_a^{b}f(x,t)dt \\ =-{\int }_a^{a+\Delta a}f(x+\Delta x,t)dt+{\int }_a^b[f(x+\Delta x,t)-f(x,t)]dt+{\int }_b^{b+\Delta b}f(x+\Delta x,t)dt \end{gathered}$$

From mean value theorem we know ${\int }_a^{b}f(t)dt=(b-a)f(\xi )$, which applies to the first and last integrals:

$$\begin{gathered} \Delta \phi =-\Delta af(x+\Delta x,{\xi }_1)+{\int }_a^b[f(x+\Delta x,t)-f(x,t)]dt+\Delta bf(x+\Delta x,{\xi }_2) \\ \frac{\Delta \phi }{\Delta x}=-\frac{\Delta a}{\Delta x}f(x+\Delta x,{\xi }_1)+{\int }_a^b\frac{f(x+\Delta x,t)-f(x,t)}{\Delta x}dt+\frac{\Delta b}{\Delta x}f(x+\Delta x,{\xi }_2) \end{gathered}$$

Now as we set $\Delta x\to 0$, we can express many of the terms as definitions of derivatives (note we pass the limit sign through the integral via bounded convergence theorem). Note now that ${\xi }_1\to a$ and ${\xi }_2\to b$, which gives us:

$$\frac{d}{dx}{\int }_a^{b}f(x,t)dt=-\frac{da}{dx}f(x,a)+{\int }_a^b\frac{\partial }{\partial x}f(x,t)dt+\frac{db}{dx}f(x+\Delta x,b)$$