3.3 The Chain Rule and Inverse Functions

1-3. Find the differential .

1.

2.

3.

4-5. Find of these functions.

4.

     Therefore,

5.

6. Differentiate .

7. Find where

     Using induction on , we get

     .

8. Find for an integer if .

      Therefore, using induction

.

9. Show that the curves and are orthogonal.

      At a point of intersection,

      .

      Thus, the intersections of the two curves are and .

      At , one derivative is not defined.

      At ,

      Therefore, the curves are orthogonal.

10-13. Find

10.

11.

12.

13.

  It is appear that the denominator terms is , and numerator is .

  Therefore

14. Find the th derivative of .

15. Prove if , then satisfies the identity .

     Therefore .

16. Use differentiation to show that

     Therefore

17-20.  Find of the following expressions.

17.

18.

19.

http://matrix.skku.ac.kr/cal-lab/cal-3-2-12.html 

1/2*(2*sqrt(x)*cos(x)/sin(x) + log(sin(x))/sqrt(x))*sin(x)^sqrt

 20.

http://matrix.skku.ac.kr/cal-lab/cal-3-2-13.html

(log(arccos(x))/(x^2+1)-arctan(x)/(sqrt(-x^2+1)*arccos(x)))*arccos(x)^arctan(x)

21. Find if .

http://matrix.skku.ac.kr/cal-lab/cal-3-2-14.html

(log(x)/(x^2 + 1) + arctan(x)/x)*x^arctan(x)

(log(x)/(x^2 + 1) + arctan(x)/x)^2*x^arctan(x) - (2*x*log(x)/(x^2 + 1)^2 + arctan(x)/x^2 - 2/((x^2 + 1)*x))*x^arctan(x)

22-23. Find and of the following functions.

22. .

     .

23. .

24. Given , show that it satisfies the following identity.

.

    Using this identity, find .

      .

      Therefore, .

                 ,

                       .

      Therefore, for is even, for is odd.

25. Given , show that .

 Since so is continuous on .

      And ,  

        .

      Therefore, .

26-28. Find of the following expressions.

26. .

      .

27. .

28. .

29. If , where and are three times differentiable, find expressions for and .

      .

      .

           .

         .

30. Given , find at the point .

   .

   .

   .

   .

   .

   Therefore,.

31. Use differentiation to show that for all .

  is no differentiation for all (Because .

32. Find an equation of the tangent line to the curve at for an arbitrary value .

  , so we have to find the tangent line at

      (Because ).

33. Establish the following derivative rules.

    (a)

    (b)

    (c)

34-46. Prove the following identities.

34.

35.

36.

37.

38.

39.

40.

 Use athematical induction, If , trivial.

      Let

       So,

41.

42.

 Let . Then , so .

      .

      Note that , but because .

      Thus, .

      So, .

43.

 Let . Then , so .

      .

     Note that , but because .

     Thus, for   So .

44.

 Let . Then

      .

45.

 Let . Then , .

      Since and ,  we have

      , so  .

46.

 Let . Then

47-50. Find

47.

48.

49.

      Therefore,

50.

     It is appear that the denominator terms is , and numerator is .

     Therefore,

51. Find the th derivative of .

52. Prove if , then satisfies the identity .

      Therefore,

53-54. Find and of the following.

53.

54.

55. Given , show that it satisfies the following identity.

    Using this identity, find .

      Therefore, .

      Therefore, for is even, for is odd.

56. Given , show that .

 Since so is continuous on .

      And   

      Therefore,

57. If , where and are three times differentiable, find expressions for and .

58. Given , find at the point .

      Therefore, .