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.

(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 .