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Chapter 3 | Derivatives
Applying the Constant Rule
Find the derivative of f ( x ) =8.
Solution This is just a one-step application of the rule:
f ′( x ) =0.
Find the derivative of g ( x ) =−3.
3.11
The Power Rule We have shown that
d dx ⎞ ⎠ =2 x and d dx ⎛ x −1/2 . At this point, you might see a pattern beginning to develop for derivatives of the form d dx ( ⎛ ⎝ x 2 ⎝ x 1/2 ⎞ ⎠ = 1 2 x n ). We continue our examination of derivative formulas by differentiating power functions of the form f ( x ) = x n where n is a positive integer. We develop formulas for derivatives of this type of function in stages, beginning with positive integer powers. Before stating and proving the general rule for derivatives of functions of this form, we take a look at a specific case, d dx ( x 3 ). Aswe go through this derivation, note that the technique used in this case is essentially the same as the technique used to prove the general case. Example 3.18 Differentiating x 3
⎛ ⎝ x 3
⎞ ⎠ .
Find d dx
Solution
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