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