Are there books that give applications of calculus in pure mathematics? Browsing different books I have come across different applications of the results of calculus. For example, in the book “Rational points on Elliptic curves”, I came across the use of derivatives to find the general form of solutions to Bachet’s equation $$y^2 - x^3 =c$$ The idea being that we graph the equation, find the slope using derivatives and then use the equation of the tangent line to conclude the general formula.
Similarly there is the famous Wallis product for $\pi$ $$\frac \pi 2 = \prod_{n=1}^\infty \frac {4n^2} {4n^2 -1}$$
And there are many results that make use of the results of calculus to come up with some amazing results.
I was wondering if someone knew of some book that takes you through some of these amazing results. Or does one have to consult books on different topics to come across these results?
 A: The applications of calculus to "pure math" are honestly innumerable. I don't know of any book which gives the kind of survey you want (it would be enormous) but there are certain subjects whose textbooks will use calculus extensively. They might be more advanced than what you're looking for, though.
One obvious field that comes to mind is differential geometry, which uses calculus to study geometry. See a list of books here for instance, as well as the relatively new Visual Differential Geometry by Needham.
Another is analytic number theory which uses techniques from calculus (often on complex numbers) to answer questions in number theory. For instance, to estimate how many primes are below some fixed number $N$.
See, for instance, any of the books here.
There's also analytic combinatorics, which uses the tools of calculus to do combinatorics and computer science. This lets us get estimates on the number of ways to tile a grid, or how quickly we would expect our code to run. See the famous book by Flajolet and Sedgewick (which is a bit dense), or Wilf's equally famous generatingfunctionology.
There are countless other subjects that use calculus for "pure math" (whatever that means), and basically any subject that starts with "differential" or "analytic" will probably do. But this should be a good starting point!

I hope this helps ^_^
