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

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

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

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

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. one-to-onenot 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.

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. But so is positive.

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

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