derivative of inverse function at point P
- Function is not one-to-one, but restricting its domain can yield a related function G which is one-to-one (and has an inverse function).
- Drag points X0 and X1 on x-axis to pick domain for one-to-one function G [plotted using black dashes].
- Green curve is reflection of black curve through the line y=x; it is the graph of inverse function [named Ginv here].
- Point P=(a,b) is on the graph of Ginv; you can move P.
- Point Q=(b,a) is mirror-image of P on graph of G.
- Tangent to G at Q is computed (using derivative of f at b) and displayed.
- Slope of tangent to G at Q is used to compute slope of tangent to at P.
- Knowing slope of tangent to Ginv at a (i.e., at point P) now lets the tangent to this inverse function to be displayed.
- Move P, see how that changes point Q and tangent there to G; also notice the two tangents (at Q and at P) meet at a point on the line y=x.
- Function f (and G) can be changed using the input box at bottom of the figure.