Illustration of the Inverse Function Theorem (Derivative of Inverse Function)

All right. Let's take a look at the tangent lines (and hence the derivatives) at these points:

Here, notice that since f and g are symmetric about the main diagonal, the same should be true of their derivatives. That is, the derivatives of the corresponding points should be reciprocal of each other — as can be confirmed below by letting the parameter 'a' runs on its own:

In other words, the derivative of a function at a point, is the reciprocal of the derivative of its inverse function at its correlate — a fact known as the Inverse Function Theorem. For more, see https://mathvault.ca/derivative-inverse-functions