# "Differentiating under the Integral sign"

**URL:** <https://boards.straightdope.com/t/differentiating-under-the-integral-sign/295032>\
**Category:** Factual Questions\
**Created:** [March 17, 2005, 7:16pm UTC](https://boards.straightdope.com/t/differentiating-under-the-integral-sign/295032 "2005-03-17T19:16:28Z")\
**Posts on this page:** 6\
**Page:** 1

<div class="post-metadata">

**Author:** ![Splanky](https://avatars.discourse-cdn.com/v4/letter/s/b2d939/32.png) [@Splanky](https://boards.straightdope.com/u/Splanky)\
**Post date:** [March 17, 2005, 7:16pm UTC](https://boards.straightdope.com/t/differentiating-under-the-integral-sign/295032/1 "2005-03-17T19:16:28Z")

</div>

In _“Surely you’re joking, Mr. Feynman!”_ by Richard Feynman, Feynman talks about how a Physics professor of his made Feynman learn Advanced Calculus by himself using a textbook. As a result, he learned a method called “differentiating parameters under the integral sign”, or just “differentiating under the integral sign” which is usually skipped over in college classes but Feynman had taught himself it anyway.

As a result, Feynman had a reputation in college because he could solve integrals that other MIT kids couldn’t, just because he learned this trick.

So, what does this phrase mean? I’ve never heard it before.

---

<div class="post-metadata">

**Author:** ![Bricker](https://avatars.discourse-cdn.com/v4/letter/b/977dab/32.png) [@Bricker](https://boards.straightdope.com/u/Bricker)\
**Post date:** [March 17, 2005, 8:17pm UTC](https://boards.straightdope.com/t/differentiating-under-the-integral-sign/295032/2 "2005-03-17T20:17:00Z")

</div>

It’s just a formula for differentiation of a definite integral whose limits are functions of the differential variable.

---

<div class="post-metadata">

**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [March 17, 2005, 8:19pm UTC](https://boards.straightdope.com/t/differentiating-under-the-integral-sign/295032/3 "2005-03-17T20:19:52Z")

</div>

I’m not sure if you’re right, **Bricker**. Check [this article](http://mathforum.org/library/drmath/view/52120.html) from Ask Dr. Math and see if that’s what you had in mind.

---

<div class="post-metadata">

**Author:** ![Bricker](https://avatars.discourse-cdn.com/v4/letter/b/977dab/32.png) [@Bricker](https://boards.straightdope.com/u/Bricker)\
**Post date:** [March 17, 2005, 8:47pm UTC](https://boards.straightdope.com/t/differentiating-under-the-integral-sign/295032/4 "2005-03-17T20:47:27Z")

</div>

> [@ultrafilter](#):
>
> I’m not sure if you’re right, **Bricker**. Check [this article](http://mathforum.org/library/drmath/view/52120.html) from Ask Dr. Math and see if that’s what you had in mind.

Not quite what I had in mind. I always thought he (Feynman) was talking about using Leibniz’s Rule of Integration.

---

<div class="post-metadata">

**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [March 17, 2005, 8:49pm UTC](https://boards.straightdope.com/t/differentiating-under-the-integral-sign/295032/5 "2005-03-17T20:49:28Z")

</div>

[Mathworld](http://mathworld.wolfram.com/LeibnizIntegralRule.html) seems to use that terminology too.

Oh well, they’re both useful tricks to know.

---

<div class="post-metadata">

**Author:** ![Hyperelastic](https://avatars.discourse-cdn.com/v4/letter/h/a3d4f5/32.png) [@Hyperelastic](https://boards.straightdope.com/u/Hyperelastic)\
**Post date:** [March 18, 2005, 12:53am UTC](https://boards.straightdope.com/t/differentiating-under-the-integral-sign/295032/6 "2005-03-18T00:53:18Z")

</div>

Leibniz rule = “differentiation under the integral sign”. In the example given in the Dr. Math link the limits of integration are constant, so you don’t have to carry the two terms that correct for the change in the limit.