2.4
The Chain Rule
Remember the composition of two functions?
f o g = f ( g ( x))
The chain rule is used when you have the
composition of two functions.
d
[f ( g ( x))]= f ' ( g ( x)) g ' ( x)
dx
Find y’ for y =
(x2
+
f(x) = x3
g(x) = x2 + 1
1)3
d
[f ( g ( x))]= f ' ( g ( x)) g ' ( x)
dx
y’ = 3(x2 + 1)2 (2x) = 6x(x2 + 1)2
Find y’ for y = (3x – 2x2)3
y’ = 3(3x – 2x2)2(3 – 4x)
Find f’(x) for f ( x) = ( x + 2)
3
(
)
2
2
(
= x +2
4x
-1 3
2 2
f ' ( x) = x + 2 (2 x) = 3 2
3 x +2
3
2
)
23
Differentiate
-7
g (t ) =
2
(2t - 3)
g’(t) = 14(2t –
rewritten as
28
=
(2t - 3) 3
3)-3(2)
Differentiate
f ( x) = x
1- x
2
2
(
rewritten as
Ê1ˆ
f ' ( x) = x Á ˜ 1 - x 2
Ë2¯
2
(
= -x 1- x
3
Factor
= -7(2t – 3)-2
)
(
x 1- x
2
)
2 12
) (- 2 x )+ (1 - x ) (2 x )
-1 2
2 -1 2
2 12
(
+ 2x 1- x
)
2 12
= x (1- x
) [-x
2 -1 2
2
+ 2(1- x
2
)]
=
(
x 2 - 3x
1- x
Differentiate
f ( x) =
x
3
x +4
2
rewritten as =
(x
x
+4
2
( x + 4) (1) - x(1 3)( x + 4)
f ' ( x) =
2
23
( x + 4)
13
2
3( x + 4) - 2 x ( x + 4)
=
3( x 2 + 4) 2 3
2
13
)
)
13
Bot * Top’ – Top * Bot’
(Bot)2
Quotient Rule
2
2
2
2
2
-2 3
-2 3
=
(2 x) Ê 3 ˆ
!Á ˜
Ë3¯
x + 12
2
(
3 x +4
2
)
43
Differentiate
Ê 3x - 1 ˆ
y=Á 2
˜
Ë x +3¯
2
Ê x 2 + 3 3 - 3x -1 2x ˆ
() (
)( ) ˜
)
Ê 3x - 1 ˆ Á (
y ' = 2Á 2
˜
2
2
Á
˜
Ë x +3¯Ë
( x + 3)
¯
1
=
2( 3x -1)( 3x 2 + 9 - 6x 2 + 2x )
=
2( 3x -1)(-3x 2 + 2x + 9)
(x
(x
2
2
+ 3)
+ 3)
3
3
Derivatives of Trigonometric Functions
d
du
[sin u ]= cos u
dx
dx
d
du
2
[tan u ]= sec u
dx
dx
d
du
2
[cot u ]= - csc u
dx
dx
d
du
[cos u ]= - sin u
dx
dx
d
du
[sec u ]= sec u tan u
dx
dx
d
du
[csc u ]= - csc u cot u
dx
dx
Applying the Chain Rule to trigonometric functions
y = sin 2x
y’ = (cos 2x) (2) = 2 cos 2x
y = cos (x – 1)
y’ = -sin (x – 1) (1) = -sin (x – 1)
y = tan 3x
y’ = sec2 3x (3) = 3 sec2 3x
y = cos (3x2)
y’ = -sin (3x2) (6x) = -6x sin (3x2)
y = cos2 3x
rewritten as y = (cos 3x)2
y’ = 2(cos 3x)1 (-sin 3x) (3)
y’ = -6 cos 3x sin 3x
Differentiate
f (t ) = sin 4t
rewritten as
1
-1 2
f ' (t ) = (sin 4t ) cos 4t (4)
2
2 cos 4t
=
sin 4t
12
(sin 4t )