In this final post in this series on inverses we consider the graphical and numerical concepts related to the derivative of the inverse and look at an important formula.
To make the notation a little less messy, let’s let g(x) = f -1(x). Then we know that f (g(x))= x. Differentiating this implicitly gives
Great formula, but one I’ve never been able to memorize and use correctly! It’s my least favorite formula, because I’m never quite sure what to substitute for what.
The graph shows a function and its inverse. It really doesn’t matter which is which, since inverse functions come in pairs: the inverse of the inverse is the original function.
Notice that the graphs are symmetric to y = x. At two points, one of which is the image of the other after reflecting over the line y = x, a tangent segment has been drawn. This segment is the hypotenuse of the “slope triangle” which is also drawn. The ratio of the vertical side of this triangle to the horizontal side is the slope (i.e. the derivative) of the tangent line.
The two triangles are congruent, so that the horizontal side of one triangle is congruent to the vertical side of the other, and vice versa. Thus the slope (the derivative) of the one tangent segment is the reciprocal of the other.
If (a, b) is a point on a function and the derivative at this point is , then the point (b, a) is on the function’s inverse and the derivative here is . This is just what my least favorite formula says: if f -1 (x) = g(x), then a = g(b) and .
What you really need to know is:
At corresponding points on a function and its inverse, the derivatives are reciprocals of each other.
This is what my least favorite formula says.
The AP exams have a clever way of testing this. (The stem may give a few more values to throw you off, or the values may be in a table.)
Given that and g is the inverse of f, Find .
The solution is reasoned this way: (5, ?) is a point on g. The corresponding point on f is (?, 5) = (2, 5). The derivative of f at this point is 3, therefore the derivative at (5, 2) on g is .