It's important to know that (in the usual setting of analysis) not every function is everywhere (or even anywhere) differentiable, but this is more orthogonal to the author's point than opposed to it. A square wave is piecewise differentiable and you can compute a piecewise derivative. The Weierstrass function is defined by an infinite series, and you can compute its derivative term-by-term by the usual rules and check that the result does not converge; it is indeed straightforward to calculate its nonexistent derivative, and this is what Weierstrass did!

In general, to even ask what it means to compute a derivative we need to specify some input language which describes functions in finite terms; we are necessarily in the world of constructions rather than (say) arbitrary set-theoretical maps between infinite sets. With this in mind, the claim that differentiation is always a straightforward computation is a strong one.