Transcript Lesson 6-2

Lesson 3-5
Chain Rule
or
U-Substitutions
Objectives
• Use the chain rule to find derivatives of
complex functions
Vocabulary
• none new
Example 1
Find the derivatives of the following:
1. g(t) = (5t³ - t + 9)4
g’(t) = 4 (5t³ - t + 9)³ (15t² - 1)
2. y = (e4x²) (cos 6x)
y’(x) = (e4x²) (-sin 6x) (6) + (8x) (e4x²) (cos 6x)
= -6 e4x² sin 6x + (8x) (e4x²) (cos 6x)
Example 2
Find the derivatives of the following:
3. d(t) = 360 + 103t – 16t²
d’(t) = 103 – 32t
d’(t³ - 2t² + 1)
d’(t³ - 2t² + 1) = 103 – 32(t³ - 2t² + 1) = -32t³ + 64t² + 71
4. f(x) = (tan x²) (sin 4x³)
f’(x) = tan x² (cos 4x³) (12x²) + (sec² x²) (2x) sin 4x³
Example 3
Find the derivatives of the following:
5. y = 62x-1
y’(x) = (ln 6) 62x-1 (2) = (2ln 6) 62x-1
6. f(x) = (tan 2x)³ (3x³ - 4x² + 7x - 9)
f’(x) = 3(tan 2x)² (2) (3x³ - 4x² + 7x - 9) + (tan 2x)³ (9x² - 8x +7)
Example 4
Assume that f(x) and g(x) are differentiable functions about
which we know information about a few discrete data
points. The information we know is summarized in the
table below:
x f(x)
-2 4
-1 3
0 -6
1
1
2 -1
f’(x)
-1
-5
-3
6
5
g(x)
5
1
8
2
1
g’(x)
6
7
-5
3
?
1. If p(x) = xf(x), find p’(2)
p’(x) = d(xf(x))/dx = (1) f(x) + f’(x) x
p’(2) = (-1) + (5) (2) = -1 + 10 = 9
2. If q(x) = 3f(x)g(x), find q’(-2)
q’(x) = d(3f(x)g(x))/dx = 3[g’(x) f(x) + f’(x) g(x)]
q’(-2) = 3[(6)(4) + (-1)(5)] = 3[24 – 5] = 57
Example 4 cont
x f(x)
-2 4
-1 3
0 -6
1
1
2 -1
f’(x)
-1
-5
-3
6
5
g(x)
5
1
8
2
1
g’(x)
6
7
-5
3
?
3. If r(x) = f(x) / (5g(x)) find r’(0)
r’(x) = d(f(x)/(5g(x)))/dx = (1/5) [g(x) f’(x) - f(x) g’(x)] / g(x)²
r’(0) = (1/5)[(8)(-3) - (-6) (-5)] / (8)²
= (1/5) [-24 -30] /64 = -54/320 = -0.16875
4. If s(x) = f(g(x)), find s’(1)
s’(x) = d(f(g(x)))/dx = f’(x) • g’(x)
s’(1) = (6) (3) = 18
[chain rule!]
Example 4 cont
x f(x)
-2 4
-1 3
0 -6
1
1
2 -1
f’(x)
-1
-5
-3
6
5
g(x)
5
1
8
2
1
g’(x)
6
7
-5
3
?
5. If t(x) = (2 – f(x)) / g(x) and t’(2) = 4, find g’(2)
t’(x) = d((2-f(x)) / g(x))/dx
= [g(x) (-f’(x)) – (2-f(x)) g’(x)] / g(x)²
t’(2) = 4 = [(1)(-5) - (2-(-1)) (x)] / (1)²
= (1/5) [-24 -30] /64 = -54/320 = -0.16875
Summary & Homework
• Summary:
– Chain rule allows derivatives of more complex
functions
– Chain rule is also known as u-substitution
• Homework:
– pg 224 - 227: 3, 4, 7, 8, 11, 14, 15, 22, 29, 32,
43, 67