1. 3.8 Derivatives of Inverse Trig Functions Lewis and Clark Caverns, Montana

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. Slopes are reciprocals. Because x and y are reversed to find the reciprocal function, the following pattern always holds: Derivative Formula for Inverses: OR The slope of an inverse f n at a point x = f(a) is equal to the reciprocal of the slope of the function at x = a

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

7. We can use implicit differentiation to find:

8. But so is positive. Recall that we restricted the domain of the function y = sinx to find y = sin -1 x Recall that cos is positive in quadrants I and IV

9. We can also findusing triangles instead… 1 x y y = sin -1 x = 1 = = = sin y = x

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

11. Some calculators contains all six inverse trig functions. The TI-83+ does not. However it is occasionally still useful to know the following:

12. Differentiate y = sin -1 (1-x 2 ) Since |x|=x if x > 0 and |x| = -x if x< 0, we could write the answer as

13. Differentiate y = cos -1 5x 7 cos y = 5x 7 = 35x 6 1 5x 7 y 