Chapter 3 | Derivatives
221
Create a table using values of x just below 3 and just above 3. x x 2 −9 x −3
2.9
5.9
2.99
5.99
2.999 5.999
3.001 6.001
3.01
6.01
3.1
6.1
After examining the table, we see that a good estimate is f ′(3) =6.
For f ( x ) = x 2 , use a table to estimate f ′(3) using Equation 3.6 .
3.2
Example 3.5 Finding a Derivative
For f ( x ) =3 x 2 −4 x +1, find f ′(2) by using Equation 3.5 .
Solution Substitute the given function and value directly into the equation. f ′( x ) = lim x →2 f ( x )− f (2) x −2
Apply the definition.
⎛ ⎝ 3 x 2 −4 x +1 ⎞
⎠ −5
f ( x ) =3 x 2 −4 x +1and f (2) =5.
= lim
Substitute
x −2
x →2
( x −2)(3 x +2) x −2
= lim = lim
Simplify and factor the numerator.
x →2
(3 x +2)
Cancel the common factor.
x →2
=8
Evaluate the limit.
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