Derivative Inverse Function Rule
Finding the Derivative of an Inverse Function
We start with some inverse function g(x) = , for which we know the formula of its inverse function, f(x).
In the app, enter the formulas for g(x) and f(x) in their input boxes. The function f is graphed in green if its checkbox is checked. The function g(x) is graphed in blue.
The geometric relationship between any pair of inverse functions is that they are reflections of each other over the line y = x. For every point (a, f -1(a)) on the graph of f -1 there is an image point (f -1(a), a) on the graph of f.
In the app we see the function f -1(x) graphed in blue if its checkbox is checked. You should see only one blue graph. If you see two different blue graphs, the the formulas entered for functions f and g are not inverses of each other. Correct this problem before proceeding, if necessary.
The value of a is controlled by the slider or input box. The point is graphed on the blue function, and its image point is graphed on the green function.
We can see the tangent lines to these two curves at these points by checking the Tangent Lines checkbox. We are wanting to find the derivative of f -1(x) at x= a. This is the slope of the blue
tangent line to the graph of f -1(x) at (a, f-1(a)) .
Notice that the reflection of this tangent line about the line y = x produces the green tangent line to the graph of f(x) at (f -1(a),a). If we have a line y = mx + b and we reflect it about the line y = x we obtain
x = my + b or y = (1/m)x - b/m. Therefore, slopes of these two tangent lines are reciprocals of each other. However, notice that the tangent line to the function f is not at an input of x = a, but rather at an input of x = f -1(a).
Therefore, we conclude that the derivative of an inverse function f -1 is the reciprocal of the derivative of the function f evaluated at f -1(x). So, if we know the formula for f '(x), then we can find the formula for the derivative of f -1.
Check the checkbox for Inverse Function Rule to see the statement of this derivative rule. Check on the checkbox for proof to see the algebraic proof of what we have seen here geometrically.