I think part of the reason Feynman got as far as he did--apart from his unusual innate talents--was his skepticism of formality.

Measure theory will tell you exactly when this result is true, but it is possible to grok the result with only a basic understanding of differentiation and integration. Feynman called this "the Babylonian approach" to mathematics.

    F(t) = integral(a,b) f(t, x) dx
         ~ sum(i) f(t, xi) * Dx

    F'(t) ~ (F(t+Dt) - F(t)) / Dt
          = integral(a,b) f(t+Dt, x)-f(t, x) dx/Dt
          ~ sum(i) (f(t+Dt, xi) - f(t, xi))/Dt Dx
          ~ sum(i) f'(t, xi) Dx
          ~ integral(a, b) f'(t, x) dx

Feynman was brilliant but I don't think his skepticism of formailty sets him aparts from other good mathematicians or even a typical good mathematics student.

https://terrytao.wordpress.com/career-advice/theres-more-to-...

Terrence Tao's measure theory text is one of the best undergraduate math textbooks I've ever read. It teaches you not just about the subject, but about how to approach it the way a mathematician does.

That is exactly why I recommended it along with his entire website. But thanks for adding the comment.

Suggested edit:

"... the way a world-class mathematician does."

(because it may have a lot to do with why his book is so much better than other texts on this topic like Rudin, Royden, and Cohn.)