13.5 The Chain Rule by SGLee - HSKim, YJLim, YKKim
1. If 
where 
, 
and 
, find 
and 
at 
and 
.
.
Calculus-Sec-13-5-Solution
13.5 The Chain Rule by SGLee - HSKim, YJLim, YKKim
1. If where , and , find and at and .
.
2. Find 
and 
if 
, where 
.
,
.
.
3. Find 
of 
.
.
4. Find 
of 
.
Take a partial derivative of both sides with respect to 
,
then we have 
.
.
5. Find 
and 
when 
is an implicitly defined function of 
and 
in
.
If 
, 
, 
.
6. Find 
, 
of 
.
because 
.
because 
.
.
7. Find 
when 
.
.
8. Let 
with 
.
(a) Show that 
and 
exist everywhere.
(b) Are 
and 
continuous at the origin? Justify your answer.
http://matrix.skku.ac.kr/cal-lab/cal-12-2-5.html
(a) If 
, 
.
Thus 
and 
exist everywhere.
is not continuous at the origin.
is not continuous at the origin.
9. A function 
is called a homogeneous function of degree 
if all the terms in 
are of degree 
.
In other words, 
for any parameter 
.
If 
is a homogeneous function of degree n then show that
This is also called Euler’s theorem for homogeneous function.
10. Verify the Euler’s theorem for the following:
(i) 
. (ii) 
.
(iii) 
. (iv) 
.
(v) 
. (vi) 
.
11. If 
is a homogeneous function of degree 
in 
, 
and 
then show that
.
12. Let 
.
Find 
at 
, 
.
13. Let 
is a homogeneous function of degree 
in 
and 
.
If 
then show that 
.
14. Let 
. Show that 
.
15. Let 
. Show that 
.
16. Show that 
is a solution of 
for all 
and 
assuming that 
is a constant.
17. If 
is the solution of the equation 
with the condition that 
as 
,
find the values of 
and 
.
18. Let 
, 
, 
and 
. Find 
.
19. Let 
, 
, 
. Find 
at 
.
