Universities Don’t Want You To Know This One Simple Trick

Contents

1 Introduction

In his book Surely You’re Joking, Mr. Feynman! (Feynman, 1992), Feynman described how he learned calculus:

I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr Bader had given me.

One day he told me to stay after class. ‘Feynman’, he said, ‘you talk too much and you make too much noise. I know why. You’re bored. So I’m going to give you a book. You go up there in the back, in the corner, and study this book, and when you know everything that’s in this book, you can talk again.’

So every physics class, I paid no attention to what was going on with Pascal’s Law, or whatever they were doing. I was up in the back with this book: Advanced Calculus, by Woods. Bader knew I had studied Calculus for the Practical Man a little bit, so he gave me the real works - it was for a junior or senior course in college. It had Fourier series, Bessel functions, determinants, elliptic functions - all kinds of wonderful stuff that I didn’t know anything about.

That book also showed how to differentiate parameters under the integral sign - it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals.

The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn’t do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.

Feynman (1992, pp. 76–77)

While I did not spend my physics classes reading calculus textbooks, Feynman’s assertion about differentiation under the integral sign not being widely taught agrees with my experience, as I also never formally learned this technique, despite coming across its application multiple times.

In the following, the Leibniz integral rule, sometimes also referred to as differentiation under the integral sign, is first stated, and subsequently two simple proofs of it are shown.

2 The Leibniz integral rule

The Leibniz integral rule states that for a function $f(x,t)$, $$\frac{d}{dx}{\int }_{a(x)}^{b(x)}f(x,t)dt={\int }_{a(x)}^{b(x)}\frac{\partial f}{\partial x}dt+f[x,b(x)]\frac{db}{dx}-f[x,a(x)]\frac{da}{dx}.$$ (1)When the integral limits $a$ and $b$ are not functions of $x$, the rule can be rewritten in a simpler form as $$\frac{d}{dx}{\int }_a^{b}f(x,t)dt={\int }_a^b\frac{\partial f}{\partial x}dt.$$ (2)

3 Proof

This section shows proofs of the Leibniz integral rule. Sec. 3.1 shows a proof of the simple form. Sec. 3.2 shows a proof of the general rule using partial differentiation, whereas Sec. 3.3 shows a proof using a forward difference. Sec. 3.2 makes use of the result proven in Sec. 3.1.

3.1 Proof of the simple form

Let $$\varphi (x)={\int }_{a(x)}^{b(x)}f(x,t)dt.$$ (3)To prove the simple form, we assume that the integral limits $a$ and $b$ are not functions of $x$. We can approximate $d\varphi /dx$ using a forward difference: $$\begin{array}{rl}\frac{\delta \varphi }{\delta x}& =\frac{1}{\delta x}\left[{\int }_a^{b}f(x+\delta x,t)dt-{\int }_a^{b}f(x,t)dt\right]\\ & ={\int }_a^b\frac{f(x+\delta x,t)-f(x,t)}{\delta x}dt.\end{array}$$ (4)In the limit, this equals $d\varphi /dx$: $$\frac{d\varphi }{dx}=\underset{\delta x\to 0}{\lim }{\int }_a^b\frac{f(x+\delta x,t)-f(x,t)}{\delta x}dt={\int }_a^b\frac{\partial f}{\partial x}dt,$$ (5)thereby proving Eqn. 2.

3.2 Proof of the general form using partial differentiation

The proof of Leibniz’s general rule in this section is very similar to the one presented by Protter & Morrey (1985, pp. 421–426).

Instead of considering $\varphi$ a function of $x$ only, we consider it a function of $x$, $a$, and $b$: $$\varphi (x)=\varphi (x,a,b).$$ (6)We begin by writing an expression for the total differential of $\varphi$: $$d\varphi ={\frac{\partial \varphi }{\partial x}|}_{a,b}dx+\frac{\partial \varphi }{\partial a}da+\frac{\partial \varphi }{\partial b}db.$$ (7)Dividing by $dx$, we obtain the following expression⁠[1][1]: The vertical line adjacent to the partial derivatives means that the quantities listed there are kept constant, in this case $a$ and $b$. One may wonder how it is possible to keep $a$ and $b$ constant while varying $x$, since $a$ and $b$ are both functions of $x$. While this is of course not possible for arbitrary $a(x)$ and $b(x)$, the phrase “kept constant” should instead be interpreted as “the effects of changes in these quantities are ignored”. The effects of changes in $a$ and $b$ are accounted for in the subsequent terms in Eqn. 7.: $$\frac{d\varphi }{dx}={\frac{\partial \varphi }{\partial x}|}_{a,b}+\frac{\partial \varphi }{\partial a}\frac{da}{dx}+\frac{\partial \varphi }{\partial b}\frac{db}{dx}.$$ (8)The first expression on the r.h.s. has already been evaluated in Sec. 3.1. The remaining two expressions will be evaluated below. Let $F$ be the indefinite integral of $f(x,t)$ w.r.t. $t$: $$F(x,t)=\int f(x,t)dt.$$ (9)Then, according to the fundamental theorem of calculus, $${\int }_a^{b}f(x,t)dt=F(x,b)-F(x,a).$$ (10)Therefore, $$\frac{\partial \varphi }{\partial a}=\frac{\partial }{\partial a}\left[F(x,b)-F(x,a)\right]=-f(x,a)$$ (11) and $$\frac{\partial \varphi }{\partial b}=\frac{\partial }{\partial a}\left[F(x,b)-F(x,a)\right]=f(x,b).$$ (12)Substituting Eqns. 2, 11, and 12 back into Eqn. 8 yields Eqn. 1.

3.3 Proof of the general form using a forward difference

The proof of Leibniz’s general rule in this section is very similar to the one presented by Woods (1926, pp. 141–144), from where Feynman learned the technique.

The differentiation of $\varphi$ will be approximated by a forward difference according to Eqn. 4. In preparation for this, we may write $$\begin{array}{rl}\varphi (x+\delta x)& ={\int }_{a+\delta a}^{b+\delta b}f(x+\delta x,t)dt\\ & ={\int }_{a+\delta a}^{a}f(x+\delta x,t)dt+{\int }_a^{b}f(x+\delta x,t)dt+{\int }_b^{b+\delta b}f(x+\delta x,t)dt,\end{array}$$ (13)where $\delta a=a(x+\delta x)-a(x)$ and $\delta b=b(x+\delta x)-b(x)$. Taking the forward difference and dividing by $\delta x$ yields δxδϕ​=δx1​{I∫a+δaa​f(x+δx,t)dt​​+II∫ab​[f(x+δx,t)−f(x,t)]dt​​+III∫bb+δb​f(x+δx,t)dt​​}[2]. (14)If $\delta x$ is sufficiently small, $f(x+\delta x)$ can be assumed to be constant between $a(x+\delta x)$ and $a(x)$, and also between $b(x)$ and $b(x+\delta x)$. Therefore, $$\begin{array}{rl}\frac{\delta \varphi }{\delta x}=& -f(x+\delta x,a)\frac{\delta a}{\delta x}\\ & +{\int }_a^b\frac{f(x+\delta x,t)-f(x,t)}{\delta x}dt\\ & +f(x+\delta x,b)\frac{\delta b}{\delta x}.\end{array}$$ (15)In the limit ($\delta x\to 0$) we obtain Eqn. 1.

References