Calculus-Sec-3-3-Solution

3.3    The Chain Rule and Inverse Functions       by SGLee - HSKim - JHLee

1-3. Find the derivative .

1.

2.

3.

4-5. Find  of these functions.

4.

, so

Therefore,

5.

.

6. Find derivative of the function .

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 ,

Product of two slopes is

Therefore, the curves are orthogonal.

At  the derivate  for the equation  does not exist (or, in other words, it is equal to infinity).

Hence the tangent line to the curve  at  is vertical.

At the same point,  the derivate  for the equation  is equal 0,

hence the tangent line for the curve   at  is horizontal.

Thus, the two curves are orthogonal at .

Find the derivatives of implicit functions

So orthogonal of two circles verified.

10-13. Find

10. ;  Note

,

, ... ,

11.

12.

13.

we observe that the denominator is , and the numerator is  .

Therefore

14. Find the th derivative of .

15. Differentiate .

By differentiating  implicitly, we have

Thus, .

16. Use differentiation to show that

Therefore

17-20.  In problems below, find .

17.

18.

21. Find  if .

22-23. Find  and  of the following functions.

22.

;

;    .  <-- 답안이 틀림  (아래 sage결과로 고쳐야함)

23.

.   <-- 답안이 틀림  (아래 sage결과로 고쳐야함)

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  for the following implicit functions.

26.

.

27.

.

28.

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

.

.

.

.

30. Given , find  at the point .

.

.

.

.

.

Therefore,.

31. Determine the points on the curve  at which the tangent is either vertical or horizontal.

```  Since dx/dy=-x^2/(2y^2), when x=0  and y!=0, a horizontal line is y=(1/2)^(1/3), and when x!=0 and y=0, a vertical line is x=1.
```

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)

(a)

(b)

34-46. Prove the following identities.

34.

.

35.

or   .

36.

or   .

37.

.

38.

.

39.

Use Mathematical Induction. If , trivial.

Let

So, .

40.

.

41.

Let . Then , so .

.

Note that , but  because .

Thus, .

So, .

42.

Let . Then , so .

.

Note that , but  because .

Thus,  for   So .

43.

Let . Then

.

44. Find , where .

.

45. Find , where .

.

46. Find , where .

.

47. Find , where .

.