# Why is differentiation under the integral sign named the Leibniz rule?

The question here asked why differentiation under the integral sign is named "Feynman's trick". That is a comparatively recent name for the method. Aside from the name "differentiation under the integral sign" for this technique, it is also called Leibniz's rule or, more precisely, the Leibniz integral rule, in many places. My question is: why is Leibniz's name attached to this result on differentiation of parametric integrals? If there was a particular result of his (something more than the Fundamental Theorem of Calculus, I presume) that is a special case of differentiation under the integral sign, I'd like to be pointed to a place where that appeared.

I have looked in several books on the history of calculus or analysis and none of them explain the reason for using Leibniz's name for differentiation under the integral sign. Cauchy, in his Résumé (1823), discussed differentiation under the integral sign in the 35th lesson here but he didn't name it after anyone.

This rule is, indeed, due to Leibniz, although it was Johann Bernoulli who realized its broader implications, and there is an interesting story to its discovery. It is told in Chapter 3 of Families of Curves and the Origins of Partial Differentiation by Engelsman. The rule appears in a Leibniz's 1697 letter to Bernoulli, as a side result in their long correspondence on the problem of orthogonal trajectories.

As originally posed in 1694, it was "Given infinitely many curves by position; find the curve that intersects them all at right angles", with the motivation that the light rays are orthogonal to the wave fronts in Huygens's wave optics. Leibniz solved the problem the same year as follows: if $$V(x,y,a)=0$$ give the family then the trajectories can be found by solving $$V_x(x,y,a)dy-V_y(x,y,a)dx=0$$. At the time, Bernoulli had only algebraic $$V$$ in mind.

In June 1696 Bernoulli posed to the readers of Acta Eruditorum his now famous brahistochrone problem. He was able to find orthogonal trajectories to their family, given by $$y=\int_{0}^x\sqrt{\frac{x}{a-x}}\,dx$$, using his optico-mechanical analogy, see Did wave optics anticipate quantum mechanics? What he pointed out in a letter to Leibniz was that his general method did not seem to work for this family, or, more generally, for families of transcendental curves given by $$y=\int_{x_0}^xp(x,a)\,dx$$. And then came the Leibniz's integral rule.

"Johann Bernoulli's great break-through for transcendental curves came in August 1697, and was an immediate consequence of Leibniz's discovery earlier that month of the interchangeability theorem for differentiation and integration. When he received Leibniz's letter containing this theorem, Bernoulli at once recognised that it opened a way to differentiation with respect to the parameter for any type of expression. There had been no difficulty interpreting $$V_a(x,y,a)$$ as far as algebraic expressions $$V(x,y,a)$$ were concerned, and now the problem of interpreting $$\frac{\partial}{\partial a}\int_{x_0}^xp(x,a)\,dx$$ had been solved as well."

I could not get hold of Leibniz's letter, but Cambridge History of Science: Volume 4, Eighteenth-Century Science, p.316 says that he used that the differential of a sum of infinitesimals is equal to the sum of their differentials. While also studying orthogonal trajectories, Euler gave a different proof in De Infinitis Curvis Eiusdem (c. 1734, published 1740), by applying antiderivatives to the equality of mixed partials.

• Thanks for the reference. The discussion of this topic occurs in Chapter 2 of that book as well, where it is pointed out that other problems about families of curves had led Leibniz to discover double integrals in 1697 too. I had not realized that multivariable calculus in both its differential and integral aspects could be traced back to Leibniz. – KCd Jan 6 at 16:41
• @KCd Newton and Waring too, arguably, see also Cajori The Early History of Partial Differential Equations and of Partial Differentiation and Integration. – Conifold Jan 7 at 7:48
• Do you know who was the first to realize that examples of definite integrals could be determined by this method? Leibniz and Bernoulli were not using differentiation under the integral sign for that purpose in their study of orthogonal trajectories as far as I could tell. In "Elements of the Integral Calculus" by Byerly (1888) there are some examples of such evaluations: see pages 96 and 105-108 (the book is online at archive.org/details/cu31924004779447/page/n117). – KCd Jan 9 at 22:48
• @KCd Sorry, I do not know off the top of my head. Abel seems to be doing something like that using "Laplace" transforms (my French is not good though), but I wouldn't be surprised if already Euler did it somewhere. – Conifold Jan 9 at 23:06