Exercises

Find the specified derivative(s) using the chain rule:
1.
 dw
dt
if w = x2y2 and
2.
 dw
dt
if w = x3y and
x = t4, y = t5
x = t, y = t2
3.
 dw
dt
if w = x2+y2 and
4.
 dw
dt
if w = x2-y2 and
x = cos( t) , y = sin( t)
x = cosh( t) , y = sinh( t)
5.
 dw
dt
if w = xy and
6.
 dw
dt
if w = x2-y2 and
x = et, y = e-t
x = sec( t) , y = tan( t)
7.
 dw
dt
if w = x2+2xy and
8.
 dw
dt
if w = x3y2 and
x = cos( t) , y = sin( t)
x = et, y = e-t
9.
 dw
dt
if w = tan-1 æ
è 		 y
x
 ö
ø 		and
10.
 dw
dt
if w = sin-1( xy) and
x = sin( t) , y = cos( t)
x = cos( t) , y = tan( t)


11.
 w
u
 ,  w
v
if w = x2+y2 and
12.
 w
u
 ,  w
v
if w = xsin( y) and
x = u2v, y = ( u+v) 3
x = 2uv, y = u2-v2
13.
 w
u
 ,  w
v
if w = sin( x) cot( y)
14.
 w
u
 ,  w
v
if w = sin( xy)exy
x = sin-1( uv) , y = tan-1( uv)
x = uv-1ln( v) , y = u-1v
15.
 w
x
 ,  w
t
if w = u2+v2 and
16.
 w
u
 ,  w
v
if w = u2+v2 and
u =  1
t

e-x2/(4t), v =  1
t
 e-x2/(2t)
u = sin( x-ct) , v = cos(x-ct)
17.
F¢( t)
if F( t) = ó
õ 		t

0 
 sin( u2t) du
18.
F¢( t)
if F( t) = ó
õ 		t

0 
 e-u2/tdu
19.
F¢( t)
if F( t) = ó
õ 		t+h

t 
 sin( u-t) dt
20.
F¢(t)
if F( t) = ó
õ 		t

0 
  e-u
u2+t2
 du

Compute Ñf and then use it to compute df/dt  using the vector form of the chain rule.
21.
f( x,y) = x2+y3
22.
f( x,y) = x3y2
r( t) = á t2,t3 ñ
r( t) = á t2,t3 ñ
23.
f( x,y) = x2+2xy
24.
f( x,y) = x3y2
r( t) = á cos( t) ,sin(t) ñ
r( t) = áet,e-t ñ
25.
f( x,y) = xcos( y)
26.
f(x,y) = tan-1 æ
è 		 y
x
 ö
ø
r( t) = á sin( t),t ñ
r( t) = á cos( t) ,sin( t) ñ
27.
f( x,y) = ó
õ 		y

x 
 e-u2du
28.
f(x,y) = ó
õ 		y

x 
 e-u2du
r( t) = á cos( t) ,sin(t) ñ
r( t) = át,t ñ

29. Compute dw/dt for w = x2-y2, x = cos( t) , y = sin( t) in two different ways:

    1. By substituting x = cos( t) , y = sin( t) into w = x2-y2, simplifying, and computing the derivative.
    2. By using the chain rule for two variables, and then simplifying.

30. Compute dw/dt for w = x3+xy2, x = cos(t) , y = sin( t) in two different ways:

    1. By substituting x = cos( t) , y = sin( t) into w = x3+xy2, simplifying, and computing the derivative.
    2. By using the chain rule for two variables, and then simplifying.

31. Prove that the derivative of a sum is the sum of the derivatives by applying the chain rule for 2 variables to
w = x+y
where x = f( t) and y = g( t) .

32. Prove that if w = f( r) where r = á g( u,v) ,h( u,v) ñ , then
 w
u
 =  df
dr
 ·  r
u
 ,         w
v
 =  df
dr
 ·  r
v
 ,

33. Prove the product rule by applying the chain rule for 2 variables to
w = xy
where x = f( t) and y = g( t) .

34. Prove the quotient rule by applying the chain rule for 2 variables to
w =  x
y
where x = f( t) and y = g( t) .

35. Suppose that K( x,u) is differentiable in x and suppose that for all e > 0, there is a interval (p,q) such that if x is in ( p,q) , then
ê
ê 		 K( x+h,u) -K( x,u)
h
 -Kx(x,u) ê
ê 		< e
(6)
independent of u. Show that
 d
dx
 ó
õ 		b

a 
 K( x,u) du = ó
õ 		b

a 
  K
x
 ( x,u) du

    1. Let f( x) = òabK( x,u) du and show that
       f( x+h) -f( x)
      h
      		 = ó
      õ       		b

      a 
      		 æ
      è       		 K( x+h,u) -K( x,u)
      h
      		 ö
      ø       		du

    2. Show that if x is in ( p,q) where ( p,q) is an interval on which (6) holds, then
      ê
      ê       		 f( x+h) -f( x)
      h
      		 - ó
      õ       		b

      a 
      		  K
      x
      		 ( x,u) du ê
      ê       		< e(b-a)
      What does this imply about  f' ( x) ?

36. This exercise uses 2 different methods to differentiate the indefinite integral
F( t) = ó
õ 		t

0 
 eu-tdu

    1. Find F¢( t) using the chain rule for functions of 2 variables.
    2. Write F( t) as the product of two functions of t and apply the product rule. Is the result the same in both cases?

37. Show that the convolution function
y( t) = ó
õ 		t

0 
 et-uf( u) du
is a solution to y' - y = f( t) .

38. Show that the convolution function
y( t) = ó
õ 		t

0 
 sin( t-u) f( u) du
is a solution to y'' + y = f( t) .

39. Implicit Differentiation: Show that if F( x,y) = k where k is constant, then
 dy
dx
 =  -Fx
Fy
(hint: use the chain rule and the fact that
 dy/dt
dx/dt
 =  dy
dx

40. Implicit Differentiation: Use implicit differentiation to compute y'  = dy/dx for
x4y+xy+xy4 = 3
Then let F( x,y) = x4y+xy+xy4 and use the formula in exercise 29.

41. Suppose that z = f( x,y) and that x = p+mt and y = q+nt, where m, n, p, and q are constants. Show that
z¢( t) = mfx+nfy        and       z¢¢( t) = fxxm2+2mnfxy+fyyn2
42. A function f( x,y) is said to be homogeneous of degree n if
f( tx,ty) = tnf( x,y)
(7)
for all real numbers t. For example, f( x,y) = x3+3xy2 is homogeneous of degree 3 since
f( tx,ty)
=
( tx) 3+3( tx) (ty) 2
=
t3x2+t33xy2
=
t3f( x,y)
Show that if a differentiable function f( x,y) is homogeneous of degree n, then
x  f
x
 +y  f
y
 = nf(x,y)
(hint: Differentiate both sides of (7) with respect to t-use the chain rule to differentiate f( tx,ty) -and then let t = 1).