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Chapter 3 | Derivatives
f ( x ). This relationship is illustrated in Figure 3.18 .
Figure 3.18 The derivative of g ( x ) is 3 times the derivative of f ( x ).
Example 3.21 Applying Basic Derivative Rules
Find the derivative of f ( x ) =2 x 5 +7.
Solution We begin by applying the rule for differentiating the sum of two functions, followed by the rules for differentiating constant multiples of functions and the rule for differentiating powers. To better understand the sequence in which the differentiation rules are applied, we use Leibniz notation throughout the solution: f ′( x ) = d dx ⎛ ⎝ 2 x 5 +7 ⎞ ⎠
⎛ ⎝ 2 x 5
⎞ ⎠ + d
= d
dx (7) Apply the sum rule.
dx
⎛ ⎝ x 5 ⎞ ⎠ + d dx (7) Apply the constant multiple rule.
=2 d
dx ⎛ ⎝ 5 x 4
⎞ ⎠ +0
=2
Apply the power rule and the constant rule.
=10 x 4 .
Simplify.
Find the derivative of f ( x ) =2 x 3 −6 x 2 +3.
3.14
Example 3.22 Finding the Equation of a Tangent Line
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