Derivatives of Inverse Trig Functions Explained: A Comprehensive Guide

1. 3.8 Derivatives of Inverse Trig Functions

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

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

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

9. We can use implicit differentiation to find:

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

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

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

13. Homework: 3.8a 3.8 p170 3,12 3.7 p162 33,42,51 2.3 p85 51 3.8b 3.8 p170 6,15,21,29 2.4 p92 5,23