The Chain Rule
Reading assignment: 2.5
Recommended exercises: 2.5: 1,5, 9,13,20,22,23,26
1. Let w  f (u , v), where u  xy and v 
w
x
w
. Using the chain rule, express
and
in terms
x
y
y
of x, y, fu , and f v .
 x  er cos 


and
2. Suppose that 
and w  f ( x, y ) . Express the differential operators
in
r

r


y

e
sin






terms of r , ,
. For fun you might try finding expressions for
in terms of
and
and
x
y
x
y
r , ,


and
.
r

3. Suppose that z  f ( x  y, x  y ) has continuous partial derivatives with respect to u  x  y
z z  z   z 
     
x y  u   v 
2
and v  x  y . Show that
2
4. Let f :

3
2
be given by f ( x1 , x2 )  ( x1  x2 , x1 x2 , x12 x2 ) and let x :
3

x (t1 , t2 , t3 )  (t1t2 , t ) .
2
1
a) Write an explicit formula for f x


b) Find D f x .
 
 
c) Find D f and D  x  and the product D f D( x ) to get your answer from b).
2
be given by
5. Let f :
2

and g :
2

 f  g  f  f g
. Prove that    
g2
g
.
 2 1
 x sin   if x  0
6. Let f ( x)  
(Why is this problem here? There is only one variable!)
 x
 0
if x = 0

a. Find f (0) if it exists.
b. Find a formula for f ( x ) for all x.
c. Is f  continuous (and so what?). Explain.