Logarithmic, Exponential, and Other Transcendental Functions 5 Copyright © Cengage Learning. All rights reserved.

Inverse Trigonometric Functions: Differentiation Copyright © Cengage Learning. All rights reserved. 5.6

3 Develop properties of the six inverse trigonometric functions. Differentiate an inverse trigonometric function. Review the basic differentiation rules for elementary functions. Objectives

4 Inverse Trigonometric Functions

5 None of the six basic trigonometric functions has an inverse function. This statement is true because all six trigonometric functions are periodic and therefore are not one-to-one. In this section you will examine these six functions to see whether their domains can be restricted in such a way that they will have inverse functions on those restricted domains.

6 D(f) = [-pi/2, pi/2] R(f) = [-1, 1] D(g) = R(f) = [-1, 1] R(g) = D(f) = [-pi/2, pi/2] y = arcsin(x) iff sin(y) = x y = arccsc(x) iff csc(y) = x D(f) = [-pi/2, pi/2] and x.ne. 0 R(f) = (-inf, -1] and [1,inf) D(g) = R(f) = |x| >= 1 R(g) = D(f) = [-pi/2, pi/2] y.ne. 0

7 D(f) = [0, pi] R(f) = [-1, 1] D(g) = R(f) = [-1, 1] R(g) = D(f) = [0, pi] y = arccos(x) iff cos(y) = x D(f) = [0, pi] x.ne. pi/2 R(f) = (-inf, -1] and [1,inf) D(g) = R(f) = |x| >= 1 R(g) = D(f) = [0, pi] y.ne. pi/2 y = arcsec(x) iff sec(y) = x

8 D(f) = [-pi/2, pi/2] R(f) = [-inf, inf] D(g) = R(f) = [-inf, inf] R(g) = D(f) = [-pi/2, pi/2] y = arctan(x) iff tan(y) = x D(f) = [0, pi] R(f) = [-inf, inf] D(g) = R(f) = [-inf, inf] R(g) = D(f) = [0, pi] y = arccot(x) iff cot(y) = x

9 Under suitable restrictions, each of the six trigonometric functions is one-to-one and so has an inverse function, as shown in the following definition. Inverse Trigonometric Functions

10 The graphs of the six inverse trigonometric functions are shown in Figure Figure 5.29 Inverse Trigonometric Functions

11 Example 1 – Evaluating Inverse Trigonometric Functions Evaluate each function. Solution:

12 cont’d Example 1 – Evaluating Inverse Trigonometric Functions My examples

13 Inverse functions just like any inverse functions have the properties f (f –1 (x)) = x and f –1 (f (x)) = x. When applying these properties to inverse trigonometric functions, remember that the trigonometric functions have inverse functions only in restricted domains. For x-values outside these domains, these two properties do NOT hold. For example, arcsin(sin π) is equal to 0, not π. Inverse Trigonometric Functions My example:

14 Example 2 – Solving an Equation My example

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16 My example Note that arcsin and arctan have range –pi/2 to pi/2. That is why we should take 1 st two triangles down to account for negtive. However, the range of arccos is 0 to pi => We have to take it to the left.

17 Derivatives of Inverse Trigonometric Functions The derivative of the transcendental function f (x) = ln x is the algebraic function f'(x) = 1/x. You will now see that the derivatives of the inverse trigonometric functions also are algebraic.

18 The following theorem lists the derivatives of the six inverse trigonometric functions. Derivatives of Inverse Trigonometric Functions

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21 The absolute value sign is not necessary because e 2x > 0. Example 4 – Differentiating Inverse Trigonometric Functions

22 My examples Note simplification in this one:

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24 max and min inflection point My example (optional):

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26 Review of Basic Differentiation Rules

27 Review of Basic Differentiation Rules An elementary function is a function from the following list or one that can be formed as the sum, product, quotient, or composition of functions in the list. With the differentiation rules introduced so far in the text, you can differentiate any elementary function. For convenience, these differentiation rules are summarized below.

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