Chapter 3 Section 07

Chapter 3 Section 07 - 3.7 Chain Rule page 171 1 Objectives...

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1 3.7 Chain Rule page 171
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2 Objectives 1. Find derivatives of “nested functions” using the chain rule. 2. Find the equation of a line tangent to a curve described by a “nested function.”
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3 Chain Rule If f and g are both differentiable functions and F = fog(x) = f(g(x)), then F is differentiable and F’ = f’(g(x))g’(x) or g(x)     u   and   f(u)     y   where = = = dx du du dy dx dy
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4 Example: f(x) = (1 + x 4 ) 2/3 f’(x) = (2/3)(1 + x 4 ) -1/3 ∙D(1+x 4 ) f’(x) = (2/3)(1 + x 4 ) -1/3 ∙4x 3
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5 Example: f(t) = 3 √(1 + tan(t)) = (1 + tan(t)) 1/3 f’(t) = (1/3)(1 + tan(t)) -2/3 ∙D(1+tan(t)) f’(t) = (1/3)(1 + tan(t)) -2/3 ∙sec 2 (t)
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6 Example: y = 4sec(5x) y’ = 4sec(5x)tan(5x)∙D(5x) y’ = 4sec(5x)tan(5x)∙5
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7 Example: y = e -5x ∙cos(3x) y’ = e -5x ∙Dcos(3x) + cos(3x)∙De -5x y’ = e -5x ∙(-sin(3x)(3)) + cos(3x)∙e -5x (-5)
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8 Example: y = sin(sin(sin(x))) y’ = cos(sin(sin(x)))∙Dsin(sin(x)) y’ = cos(sin(sin(x)))∙cos(sin(x))∙Dsin(x) y’ = cos(sin(sin(x)))∙cos(sin(x))∙cos(x)
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9 Example 2 3 2 3 2 3 4 3 3 3 2 3 2 3 2 3 4 3 3 3 3 3 4 3 3 3 4 1 3 3 4 3 3 1 3 1 3 1 1 1 4 1 1 3 1 3 1
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Chapter 3 Section 07 - 3.7 Chain Rule page 171 1 Objectives...

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