Question

Getting Started 1. State the derivatives of sin^{-1} x, tan^{-1} x, and sec^{-1} x. 2. Find the slope of the line tangent to the graph of y = sin^{-1} x at x = 0. 3. Find the slope of the line tangent to the graph of y = tan^{-1} x at x = -2. 4. How are the derivatives of sin^{-1} x and cos^{-1} x related? 5. Suppose f is a one-to-one function with f(2) = 8 and f'(2) = 4. What is the value of (f^{-1})'(8)? 6. Explain how to find (f^{-1})'(y_0), given that y_0 = f(x_0). 7–8. Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. 7. x: -3 -2 -1 0 1 2 f(x): 2 3 4 6 7 f'(x): 1/2 2 3/2 ? a. (f^{-1})'(4) b. (f^{-1})'(6) c. (f^{-1})'(1) d. f'(1) 8. x: -4 -2 0 2 4 f(x): 0 1 2 3 4 f'(x): 5 4 3 2 1 a. f'(f(0)) b. (f^{-1})'(0) c. (f^{-1})'(1) d. (f^{-1})'(4) 9. If f is a one-to-one function with f(3) = 8 and f'(3) = 7, find the equation of the line tangent to y = f^{-1}(x) at x = 8. 10. The line tangent to the graph of the one-to-one function y = f(x) at x = 3 is y = 5x + 1. Find f'(16) and (f^{-1})'(16). 11. Find the slope of the curve y = sin^{-1} x at (1/2, ?/6) without calculating the derivative of sin^{-1} x. 12. Find the slope of the curve y = tan^{-1} x at (1, ?/4) without calculating the derivative of tan^{-1} x. Practice Exercises 13–40. Evaluate the derivative of the following functions. 13. f(x) = sin^{-1}(2x) 14. f(x) = x sin^{-1} x 15. f(x) = cos(sin^{-1} 2x) 16. f(x) = sin^{-1}(ln x) 17. f(x) = sin^{-1}(e^{-x}) 18. f(x) = sin^{-1}(sin x) 19. f(x) = tan^{-1}(10x) 20. f(x) = 2x tan^{-1} x - ln(1 + x^2) 21. f(x) = tan^{-1}(2x^2 - 4) 22. g(x) = tan^{-1}(1/x) 23. f(x) = cot^{-1}(?x) 24. f(x) = sec^{-1}(?x) 25. f(x) = x^2 + 2x^3 cot^{-1} x - ln(1 + x^2) 26. f(x) = x cos^{-1} x - ?(1 - x^2)

          Getting Started

1. State the derivatives of sin^{-1} x, tan^{-1} x, and sec^{-1} x.

2. Find the slope of the line tangent to the graph of y = sin^{-1} x at x = 0.

3. Find the slope of the line tangent to the graph of y = tan^{-1} x at x = -2.

4. How are the derivatives of sin^{-1} x and cos^{-1} x related?

5. Suppose f is a one-to-one function with f(2) = 8 and f'(2) = 4. What is the value of (f^{-1})'(8)?

6. Explain how to find (f^{-1})'(y_0), given that y_0 = f(x_0).

7–8. Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined.

7. x: -3 -2 -1 0 1 2
   f(x): 2 3 4 6 7
   f'(x): 1/2 2 3/2 ?
   a. (f^{-1})'(4)   b. (f^{-1})'(6)   c. (f^{-1})'(1)   d. f'(1)

8. x: -4 -2 0 2 4
   f(x): 0 1 2 3 4
   f'(x): 5 4 3 2 1
   a. f'(f(0))   b. (f^{-1})'(0)   c. (f^{-1})'(1)   d. (f^{-1})'(4)

9. If f is a one-to-one function with f(3) = 8 and f'(3) = 7, find the equation of the line tangent to y = f^{-1}(x) at x = 8.

10. The line tangent to the graph of the one-to-one function y = f(x) at x = 3 is y = 5x + 1. Find f'(16) and (f^{-1})'(16).

11. Find the slope of the curve y = sin^{-1} x at (1/2, ?/6) without calculating the derivative of sin^{-1} x.

12. Find the slope of the curve y = tan^{-1} x at (1, ?/4) without calculating the derivative of tan^{-1} x.

Practice Exercises

13–40. Evaluate the derivative of the following functions.

13. f(x) = sin^{-1}(2x)
14. f(x) = x sin^{-1} x
15. f(x) = cos(sin^{-1} 2x)
16. f(x) = sin^{-1}(ln x)
17. f(x) = sin^{-1}(e^{-x})
18. f(x) = sin^{-1}(sin x)
19. f(x) = tan^{-1}(10x)
20. f(x) = 2x tan^{-1} x - ln(1 + x^2)
21. f(x) = tan^{-1}(2x^2 - 4)
22. g(x) = tan^{-1}(1/x)
23. f(x) = cot^{-1}(?x)
24. f(x) = sec^{-1}(?x)
25. f(x) = x^2 + 2x^3 cot^{-1} x - ln(1 + x^2)
26. f(x) = x cos^{-1} x - ?(1 - x^2)

Getting Started

1. State the derivatives of sin^-1 x, tan^-1 x, and sec^-1 x.

2. Find the slope of the line tangent to the graph of y = sin^-1 x at x = 0.

3. Find the slope of the line tangent to the graph of y = tan^-1 x at x = -2.

4. How are the derivatives of sin^-1 x and cos^-1 x related?

5. Suppose f is a one-to-one function with f(2) = 8 and f'(2) = 4. What is the value of (f^-1)'(8)?

6. Explain how to find (f^-1)'(y0), given that y0 = f(x0).

7–8. Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined.

7. x: -3 -2 -1 0 1 2
f(x): 2 3 4 6 7
f'(x): 1/2 2 3/2 ?
a. (f^-1)'(4) b. (f^-1)'(6) c. (f^-1)'(1) d. f'(1)

8. x: -4 -2 0 2 4
f(x): 0 1 2 3 4
f'(x): 5 4 3 2 1
a. f'(f(0)) b. (f^-1)'(0) c. (f^-1)'(1) d. (f^-1)'(4)

9. If f is a one-to-one function with f(3) = 8 and f'(3) = 7, find the equation of the line tangent to y = f^-1(x) at x = 8.

10. The line tangent to the graph of the one-to-one function y = f(x) at x = 3 is y = 5x + 1. Find f'(16) and (f^-1)'(16).

11. Find the slope of the curve y = sin^-1 x at (1/2, ?/6) without calculating the derivative of sin^-1 x.

12. Find the slope of the curve y = tan^-1 x at (1, ?/4) without calculating the derivative of tan^-1 x.

Practice Exercises

13–40. Evaluate the derivative of the following functions.

13. f(x) = sin^-1(2x)
14. f(x) = x sin^-1 x
15. f(x) = cos(sin^-1 2x)
16. f(x) = sin^-1(ln x)
17. f(x) = sin^-1(e^-x)
18. f(x) = sin^-1(sin x)
19. f(x) = tan^-1(10x)
20. f(x) = 2x tan^-1 x - ln(1 + x^2)
21. f(x) = tan^-1(2x^2 - 4)
22. g(x) = tan^-1(1/x)
23. f(x) = cot^-1(?x)
24. f(x) = sec^-1(?x)
25. f(x) = x^2 + 2x^3 cot^-1 x - ln(1 + x^2)
26. f(x) = x cos^-1 x - ?(1 - x^2)

Added by Mercedes J.

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