Chain rule

1. THE CHAIN RULE (CASE 1) Suppose that
z=f (x, y) is a differentiable function of x and y,
where x=g (t) and y=h (t) and are both
differentiable functions of t. Then z is a
differentiable function of t and
dt
dy
y
f
dt
dx
x
f
dt
dz
∂
∂
+
∂
∂
=
dt
dy
y
z
dt
dx
x
z
dt
dz
∂
∂
+
∂
∂
=

Question No. 1.
Use THE CHAIN RULE to find the derivative of
w = xy with respect to t along the path x = cost
and y = sint. What is the derivative’s value at t
= Pie/2.
).........(i
dt
dy
y
w
dt
dx
x
w
dt
dw
∂
∂
+
∂
∂
=
Solution:
We Use THE CHAIN RULE to find the
derivative of w = xy with respect to t along the
path x = cost and y = sint as follows:
x
y
xy
dy
dw
andy
x
xy
dx
dw
=
∂
∂
==
∂
∂
=
)()(

t
dt
td
dt
dy
andt
dt
td
dt
dx
cos
sin
sin
cos
==−==
Solution continued……….
ttt
tttt
txty
dt
dy
y
w
dt
dx
x
w
dt
dw
ieqnfromThen
2cossincos
).(coscos)sin.(sin
)(cos)sin(
)(
22
=−=
+−=
+−=
∂
∂
+
∂
∂
=
1cos
2
.2cos
2/
−==
=
=
π
π
πt
dt
dw
partFurther

2. THE CHAIN RULE (CASE 2) Suppose that
w=f (x, y, z) is a differentiable function of x, y
and z and are all differentiable functions of t.
Then w is a differentiable function of t and
dt
dz
z
f
dt
dy
y
f
dt
dx
x
f
dt
dw
∂
∂
+
∂
∂
+
∂
∂
=
dt
dz
z
w
dt
dy
y
w
dt
dx
x
w
dt
dw
∂
∂
+
∂
∂
+
∂
∂
=

Question No. 2.
Use THE CHAIN RULE to find the derivative of
w = xy + z with respect to t along the path x =
cost , y = sint and z = t. What is the derivative’s
value at t = 0.
).........(i
dt
dz
z
w
dt
dy
y
w
dt
dx
x
w
dt
dw
∂
∂
+
∂
∂
+
∂
∂
=
Solution:
We Use THE CHAIN RULE to find the
derivative of w = xy + z with respect to t along
the path x = cost, y = sint and z = t as follows:
1
)()(
,
)(
=
∂
+∂
==
∂
+∂
==
∂
+∂
=
z
zxy
dz
dw
andx
y
zxy
dy
dw
andy
x
zxy
dx
dw

1cos
sin
,sin
cos
====−==
dt
dt
dt
dz
andt
dt
td
dt
dy
t
dt
td
dt
dx
Solution continued……….
ttt
tttt
txty
dt
dz
z
w
dt
dy
y
w
dt
dx
x
w
dt
dw
ieqnfromThen
2cos11sincos
1).(coscos)sin.(sin
1.1)(cos)sin(
)(
22
+=+−=
++−=
++−=
∂
∂
+
∂
∂
+
∂
∂
=
211
0.2cos1
0
=+=
+=
=t
dt
dw
partFurther

3. THE CHAIN RULE (CASE 3) Suppose that z=f
(x, y) is a differentiable function of x and y, where
x=g (s, t) and y=h (s, t) are differentiable
functions of s and t. Then
ds
dy
y
z
ds
dx
x
z
dx
dz
∂
∂
+
∂
∂
=
dt
dy
y
z
dt
dx
x
z
dt
dz
∂
∂
+
∂
∂
=

4. THE CHAIN RULE (GENERAL VERSION)
Suppose that u is a differentiable function of the n
variables x1, x2,‧‧‧,xn and each xj is a differentiable
function of the m variables t1, t2,‧‧‧,tm Then u is a
function of t1, t2,‧‧‧, tm and
for each i=1,2,‧‧‧,m.
i
n
niii t
x
x
u
‧‧‧
dt
x
x
u
dt
dx
x
u
t
u
∂
∂
∂
∂
++
∂
∂
∂
+
∂
∂
=
∂
∂ 2
2
1
1

F (x, y)=0. Since both x and y are functions of
x, we obtain
But dx /dx=1, so if ∂F/∂y≠0 we solve for
dy/dx and obtain
0=
∂
∂
+
∂
∂
dx
dy
y
F
dx
dx
x
F
y
x
F
F
y
F
x
F
dx
dy
−=
∂
∂
∂
∂
−=

F (x, y, z)=0
But and
so this equation becomes
If ∂F/∂z≠0 ,we solve for ∂z/∂x and obtain the
first formula in Equations 7. The formula for
∂z/∂y is obtained in a similar manner.
0=
∂
∂
∂
∂
+
∂
∂
+
∂
∂
x
z
z
F
dx
dy
y
F
dx
dx
x
F
1)( =
∂
∂
x
x
1)( =
∂
∂
y
x
0=
∂
∂
∂
∂
+
∂
∂
x
z
z
F
x
F
z
F
x
F
dx
dz
∂
∂
∂
∂
−=
z
F
y
F
dy
dz
∂
∂
∂
∂
−=

1. DEFINITION A function of two variables
has a local maximum at (a, b) if f (x, y) ≤ f
(a, b) when (x, y) is near (a, b). [This means
that
f (x, y) ≤ f (a, b) for all points (x, y) in some
disk with center (a, b).] The number f (a, b) is
called a local maximum value. If f (x, y) ≥ f
(a, b) when (x, y) is near (a, b), then f (a, b) is
a local minimum value.

2. THEOREM If f has a local maximum or
minimum at (a, b) and the first order partial
derivatives of f exist there, then fx(a, b)=1 and
fy(a, b)=0.

A point (a, b) is called a critical point (or
stationary point) of f if fx (a, b)=0 and fy (a,
b)=0, or if one of these partial derivatives does
not exist.

3. SECOND DERIVATIVES TEST Suppose the
second partial derivatives of f are continuous
on a disk with center (a, b) , and suppose that
fx (a, b) and fy (a, b)=0 [that is, (a, b) is a critical
point of f]. Let
(a)If D>0 and fxx (a, b)>0 , then f (a, b) is a local
minimum.
(b)If D>0 and fxx (a, b)<0, then f (a, b) is a local
maximum.
(c) If D<0, then f (a, b) is not a local maximum
or minimum.
2
)],([),(),(),( bafbafbafbaDD xyyyxx −==

NOTE 1 In case (c) the point (a, b) is called a
saddle point of f and the graph of f crosses its
tangent plane at (a, b).
NOTE 2 If D=0, the test gives no information:
f could have a local maximum or local
minimum at (a, b), or (a, b) could be a saddle
point of f.
NOTE 3 To remember the formula for D it’s
helpful to write it as a determinant:
2
)( xyyyxx
yyyx
xyxy
fff
ff
ff
D −==

4. EXTREME VALUE THEOREM FOR
FUNCTIONS OF TWO VARIABLES If f is
continuous on a closed, bounded set D in R2
,
then f attains an absolute maximum value
f(x1,y1) and an absolute minimum value f(x2,y2)
at some points (x1,y1) and (x2,y2) in D.

5. To find the absolute maximum and minimum
values of a continuous function f on a closed,
bounded set D:
1. Find the values of f at the critical points of in D.
2. Find the extreme values of f on the boundary of D.
3. The largest of the values from steps 1 and 2 is the
absolute maximum value; the smallest of these
values is the absolute minimum value.

Chain rule

More Related Content

What's hot

Viewers also liked

Similar to Chain rule

Recently uploaded

Chain rule