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Objectives:1. Be able to find the derivative of functions by applying
the Product Rule.
Critical Vocabulary:Derivative, Tangent
Daily Warm-Up: Find the derivative of the following functions
1. h(x) = (3x - 2x2)(5 + 4x)2. h(x) = x2(3x - 2)(x2 - 4x)
h’(x) = -24x2 + 4x + 15h’(x) = 15x4 - 56x3 + 24x2
I. The Product Rule
The Product Rule is used when two or more functions are being multiplied together. h(x) = f(x) • g(x)
Look back at our warm-up problem: f(x) = (3x - 2x2)(5 + 4x)This could be done using the PRODUCT RULE!!!!!
)(')()(')()]()([ xfxgxgxfxgxfdx
d
Example 1: Find the derivative: h(x) = (3x - 2x2)(5 + 4x)
f(x) = 3x – 2x2
f’(x) = 3 – 4x
g(x) = 5 + 4x g’(x) = 4
h’(x) = (3x - 2x2)4 + (5 + 4x)(3 - 4x)h’(x) = 12x - 8x2 + 15 - 8x - 16x2
h’(x) = -24x2 + 4x + 15
I. The Product Rule
)(')()(')()]()([ xfxgxgxfxgxfdx
d
Example 2: Use the function to find the slope of the tangent line at the point (2, 3/2)
f(x) = x-1 + 1 f’(x) = -x-
2 g(x) = x - 1 g’(x) = 1
h’(x) = (x-1 + 1 )1 + (x - 1)(-x-2)h’(x) = x-1 + 1 - x-1 + x-2
111
)(
xx
xh
2
11)('x
xh
2
2 1)('
x
xxh
2
2
)2(
1)2()2('
h
4
14)2('
h
4
5)2(' h
f(x) = 5/4x - 1
I. The Product Rule
Example 2: Use the function to find the slope of the tangent line at the point (2, 3/2)
111
)(
xx
xh
2
2 1)('
x
xxh
4
5)2(' h
f(x) = 5/4x - 1
)(')()(')()]()([ xfxgxgxfxgxfdx
d
I. The Product Rule
Example 3: Use the function to find the equation of the tangent line at the point (-1, 4): h(x) = 2x(x2 + 3x) f(x) =
2x f’(x) = 2 g(x) = x2 +
3x g’(x) = 2x + 3
h’(x) = (2x)(2x + 3) + (x2 + 3x)(2)h’(x) = 4x2 + 6x + 2x2 + 6xh’(x) = 6x2 + 12xh’(-1) = 6(-1)2 + 12(-1)
h’(-1) = 6 - 12h’(-1) = -6
4 = -6(-1) + b4 = 6 + b
-2 = bf(x) = -6x - 2
)(')()(')()]()([ xfxgxgxfxgxfdx
d
I. The Product Rule
Example 3: Use the function to find the equation of the tangent line at the point (-1, 4): h(x) = 2x(x2 + 3x)
h’(x) = 6x2 + 12xh’(-1) = -6
f(x) = -6x - 2
)(')()(')()]()([ xfxgxgxfxgxfdx
d
I. The Product Rule
The Product Rule can be extended to more than 2 functions.
)()()()( xhxgxfxj
Example 4: Find the derivative: j(x) = x2(3x - 2)(x2 - 4x )
f(x) = x2 f’(x) = 2x
g(x) = 3x - 2 g’(x) = 3
h’(x) = 2x(3x - 2)(x2 - 4x) + x2(3)(x2 - 4x) + x2(3x-2)(2x-4) h’(x) =6x4 - 28x3 + 16x2 + 3x4 - 12x3 + 6x4 – 16x3 + 8x2
h’(x) = 15x4 - 56x3 + 24x2
)(')()()()(')()()()(')(' xhxgxfxhxgxfxhxgxfxj
h(x) = x2 – 4x h’(x) = 2x - 4
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