7 INVERSE FUNCTIONS INVERSE FUNCTIONS 7 6 Inverse

7 INVERSE FUNCTIONS

INVERSE FUNCTIONS 7. 6 Inverse Trigonometric Functions In this section, we will learn about: Inverse trigonometric functions and their derivatives.

INVERSE SINE FUNCTIONS Evaluate: a. b. Example 1

INVERSE SINE FUNCTIONS Example 1 a We have: § This is because and lies between , and .

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

INVERSE SINE FUNCTIONS Example 1 b § This enables us to read from the triangle that:

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

INVERSE SINE FUNCTIONS Example 2 b Combining Formula 3 with the Chain Rule, we have:

INVERSE COSINE FUNCTIONS Formula 6 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 TANGENT FUNCTIONS 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 Equation 9 Differentiating that latter equation implicitly with respect to x, we have: Thus,

DERIVATIVES Table 11

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

DERIVATIVES Differentiate: Example 5

DERIVATIVES Example 5 a

DERIVATIVES Example 5 b