1. 7 INVERSE FUNCTIONS

2. 7.6 Inverse Trigonometric Functions In this section, we will learn about: Inverse trigonometric functions and their derivatives. INVERSE FUNCTIONS

3. Evaluate: a. b. INVERSE SINE FUNCTIONSExample 1

4. We have:  This is because, and lies between and. Example 1 aINVERSE SINE FUNCTIONS

5. Let, so.  Then, we can draw a right triangle with angle θ.  So, we deduce from the Pythagorean Theorem that the third side has length. Example 1 bINVERSE SINE FUNCTIONS

6.  This enables us to read from the triangle that: INVERSE SINE FUNCTIONSExample 1b

7. If f(x) = sin -1 (x 2 – 1), find: (a) the domain of f. (b) f ’(x). (c) the domain of f ’. INVERSE SINE FUNCTIONSExample 2

8. Combining Formula 3 with the Chain Rule, we have: Example 2 bINVERSE SINE FUNCTIONS

9. Its derivative is given by:  The formula can be proved by the same method as for Formula 3.  It is left as Exercise 17. INVERSE COSINE FUNCTIONSFormula 6

10. Since tan is differentiable, tan -1 is also differentiable. To find its derivative, let y = tan -1 x.  Then, tan y = x. INVERSE TANGENT FUNCTIONS

11. Differentiating that latter equation implicitly with respect to x, we have: Thus, INVERSE TANGENT FUNCTIONSEquation 9

12. Table 11DERIVATIVES

13. Each of these formulas can be combined with the Chain Rule.  For instance, if u is a differentiable function of x, then DERIVATIVES

14. Differentiate: DERIVATIVESExample 5

15. DERIVATIVESExample 5 a

16. DERIVATIVESExample 5 b