I was politely reminded that the proof of the chain rule has a flaw. If we have that q(x) and q(x0) are the same, we are dividing by 0. Fortunately, if we assume to know that p(q(x)) is differentiable (and ofcourse p, q are differentiable), we can pick any sequence of x's that goes to x0 in the limit, so we can avoid all such x that give q(x) = q(x0). This works in all cases, but one - if q is constant on some neighbourhood of x0. In that case (as the limit only cares about any neighbourhood of x0), we can replace q inside the limit with the constant and get 0 immediately by derivative's definition. This again proves the formula correct.