3.3 The Chain Rule and Inverse Functions by SGLee - HSKim - JHLee
1-3. Find the derivative 
.
1. 
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. 
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 
.
.
19. 
21. Find 
if 
.
