1. 3.8 Derivatives of Inverse Trig Functions
Lewis and Clark Caverns, Montana
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 1993

2. At x = 2:
We can find the inverse function as follows:
To find the derivative of the inverse function:
Switch x and y.

3. Slopes are reciprocals.
At x = 2:
At x = 4:

4. Slopes are reciprocals.
Because x and y are reversed to find the reciprocal function, the following pattern always holds:
The derivative of
Derivative Formula for Inverses:
evaluated at
is equal to the reciprocal of
the derivative of
evaluated at

5. A typical problem using this formula might look like this:
Given:
Find:
Derivative Formula for Inverses:

6. A function has an inverse only if it is one-to-one.
We remember that the graph of a one-to-one function passes the horizontal line test as well as the vertical line test.
We notice that if a graph fails the horizontal line test, it must have at least one point on the graph where the slope is zero.
one-to-one
not one-to-one

7. Now that we know that we can use the derivative to find the slope of a function, this observation leads to the following theorem:
Derivatives of Inverse functions:
If f is differentiable at every point of an interval I and df/dx is never zero on I, then f has an inverse and f -1 is differentiable at every point of the interval f(I).

8. Example: Does have an inverse?
Since is never zero, must pass the horizontal line test, so it must have an inverse.

9. We can use implicit differentiation to find:

10. We can use implicit differentiation to find:
But
so is positive.

11. We could use the same technique to find and. 1
sec d x dx -

12. p Your calculator contains all six inverse trig functions.
However it is occasionally still useful to know the following: p