Calculus BL92 Derivatives of Inverse

1
Calculus B L92 Derivatives of
Inverse Functions
Recall
Graphs of functions pass the vertical line test.
Graphs of one-to-one functions pass the
horizontal line test.
One-to-one functions have inverses that are
functions.
The graphs of inverse relations are reflections
in the line y x.
If f(x) contains (x,y), f-1(x) contains (y,x).
2
Tangent lines that pass through mirror
image points of inverse functions have
slopes that are reciprocals.
slope 3
3
Derivative of Inverse
f(-1) 3
4
Ex1. Suppose that f(x) x3 x 1. Find
the slope of the graph of the inverse of f
at the point (-1,0)
f(x) 3x2 1
5
Ex2. Suppose f(x) x3 2x and h is the
inverse of f. Evaluate h(3) and h(3).
y x3 2x
x y3 2y
x h3 2h
h(3)
3 h3 2h
h(3) 1
f(x) 3x2 2