Let f and g be inverse functions in a neighborhood about x. By inverse, we don't mean reciprocal; we mean g(f(x)) = x, and f(g(y)) = y. Examples are square and square root.
Let f have a nonzero derivative at a point s inside this inverse neighborhood. We would like to know the derivative of g at t, where t = f(s). Write the standard difference quotient in a slightly differen form:
g(y)-g(t) over y-t, as y approaches t
Limit composition is another theorem that we will accept as valid, so replace y with f(x) and let x approach s. The top becomes x-s and the bottom becomes f(x)-f(s). This is the reciprocal of a limit that is known to exist, and is nonzero, namely f′(s). Thus g and f have reciprocal derivatives.
This is clear if you look at the geometry. Reflect a graph through the main diagonal to get the inverse function. Look at the line tangent to the graph at the point of differentiation. After reflection, the slope of the line is inverted, and so is the derivative. If the tangent line was horizontal, it is vertical in the reflected graph. Thus a zero derivative leads to an infinite (ill-defined) derivative in the inverse function.