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DifferentiationTechniques:
ThePowerandSum-DifferenceRules
1.5
OBJECTIVE
• Differentiate using the Power Rule or the Sum-Difference Rule.
• Differentiate a constant or a constant times a function.
• Determine points at which a tangent line has a specified slope.
2012 Pearson Education, Inc.
All rights reserved Slide 1.51-2
1.5DifferentiationTechniques: ThePowerRule andSum-DifferenceRules
Leibniz’s Notation:
When y is a function of x, we will also designate the derivative, f¢(x), as
dy dx
which is read “the derivative of y with respect to x.”
2012 Pearson Education, Inc. All rights reserved Slide 1.5-3
1.5DifferentiationTechniques: ThePowerRule andSum-DifferenceRules
THEOREM 1: The Power Rule
For any real number k,
dy k
dx
= k×xk 1
2012 Pearson Education, Inc. All rights reserved Slide 1.5-4
1.5DifferentiationTechniques: ThePowerRule andSum-DifferenceRules
Example 1: Differentiate each of the following:
a) y = x5 b) y = x c) y = x 4
a)dx x5 d 5
dx
= 5 x5 1 b)
= 5x4
d dx
dx x
= 1 x1 1 c)
= 1
d 4 dx
dx x 4
= 4x 4 1
= 4x 5
2012 Pearson Education, Inc. All rights reserved Slide 1.5-5
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