Good numerical integration is easy, because summing smooths out noise. Good numerical differentiation is hard, because noise is amplified.

Conversely, good symbolic integration is hard, because you can get stuck and have to try another route through a combinatoric maze. Good symbolic differentiation is easy, because just applying the next obvious operation usually converges.

Huh.

Mandatory XKCD: [1]

[1] https://xkcd.com/2117/

That's exactly right. A couple more things:

- Differenting a function composed of simpler pieces always "converges" (the process terminates). One just applies the chain rule. Among other things, this is why automatic differentiation is a thing.

- If you have an analytic function (a function expressible locally as a power series), a surprisingly useful trick is to turn differentiation into integration via the Cauchy integral formula. Provided a good contour can be found, this gives a nice way to evaluate derivatives numerically.