The Chain Rule
Lesson 4.4
Derivatives of Composite Functions
How do you differentiate a composite (i.e. f (g(x)) ) function?
The Chain Rule – If f (u) is a differentiable function of u
and u = g(x) is a differentiable function of x, then f(g(x)) is a
differentiable function of x, and
d
[ f ( g ( x))]  f ' ( g ( x)) g ' ( x)
dx
Ex: Find f’(x) when f (x) =√(x2 + 3).
Let u = x2 + 3 = g(x) , so g’(x) = (2)(x2-1) + d/dx (3) = 2x
1
1
½-1)
=
u
,
so
f
’(
u)
=
(½)(u
Then f(u) = √u
=
=
½
2√u
2√x2 + 3
So, f ’(x) = f’(g(x))g’(x)
1
2x
x
(2x) =
=
=
2√x2 + 3
2√x2 + 3
√x2 + 3
More Chain Rule Examples
Find f’(x) when f (x) =(3x – 2x2)3.
Let u = 3x – 2x2 , so u’ = 3(1)x 1-1 – 2(2)x2-1 = 3 – 4x
Then f(u) = u 3 , so f ’(u) = (3)(u3-1) = 3u2
So, f ’(x) = f ’(u)u’ = 3u2 (3 – 4x) = 3(3x – 2x2)2 (3 – 4x)
3
Find all points on the graph of f (x) = (x2 – 1)2 for which f ’(x) = 0
and those for which f ’(x) does not exist.
2
3
2
2
2
f (x)= (x - 1) = (x - 1)3
= 2x
Let u = x2 – 1 , so u’ = (2)x 2-1 –
d/dx(1)
Then f(u) = u ⅔ , so f ’(u) = (⅔)(u⅔ -1) = 2
3u⅓
2
4x
4x
So, f ’(x) = f
=
(2x) =
=
3
⅓
⅓
’(u)u’
3u
3u
3 (x2 – 1)
If f ’(x) = 0, then x = 0
and f ’(x) cannot exist if x = ±1
Chain Rule & Product Rule Together
Ex: Find k’(x) when k(x) =x2√(1 – x2).
Product Rule
d
[ f ( x) g ( x)]  f ' ( x) g ( x)  g ' ( x) f ( x)
dx
Let f(x) = x2 , so f’(x) = (2)(x2-1) = 2x
-x
2
And g(x) = √(1 – x ) , so g’(x) … =
√(1-x2)
Chain Rule
y '  f ' (u )  u '
Let u = 1 – x , then u’ = -2x
1
And f(u) = √u , then f ’(u) =
2√u
1
And so y’ =
(-2x)
2√u
-2x
-x
=
=
2√(1√(1-x2)
2
2
Chain Rule & Product Rule Together
Ex: Find k’(x) when k(x) =x2√(1 – x2).
Product Rule
d
[ f ( x) g ( x)]  f ' ( x) g ( x)  g ' ( x) f ( x)
dx
Let f(x) = x2 , so f’(x) = (2)(x2-1) = 2x
-x
2
And g(x) = √(1 – x ) , so g’(x) … =
√(1-x2)
So, k’(x) = f’(x)g(x) + g’(x)f(x)
-x
(x2)
= (2x)√(1 – x2) +
√(1-x2)
√(1-x2)
-x3
2
= (2x)√(1 – x )
+
2
√(1-x )
√(1-x2)
x
=
[2√(1 – x2)√(1-x2) + -x2] =
√(1-x2)
=
2x - 3x2
√(1 - x2)
2x(1 - x2) – x3
√(1 - x2)
Lesson Practice
1) Find f ’(x) if f (x) = (6x – 5)3
Exercises
Pg. 289 (7, 13, 18, 25, 29, 35, 41, 54, 60, 61, 63)