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Chapter 3 | Derivatives
Figure 3.7 The line is tangent to f ( x ) at x =2.
Find the slope of the line tangent to the graph of f ( x ) = x at x =4.
3.1
The Derivative of a Function at a Point The type of limit we compute in order to find the slope of the line tangent to a function at a point occurs in many applications across many disciplines. These applications include velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology. This limit occurs so frequently that we give this value a special name: the derivative . The process of finding a derivative is called differentiation . Definition Let f ( x ) be a function defined in an open interval containing a . The derivative of the function f ( x ) at a , denoted by f ′( a ), is defined by (3.5) f ′( a ) = lim x → a f ( x )− f ( a ) x − a provided this limit exists. Alternatively, we may also define the derivative of f ( x ) at a as (3.6) f ′( a ) = lim h →0 f ( a + h )− f ( a ) h .
Example 3.4 Estimating a Derivative
For f ( x ) = x 2 , use a table to estimate f ′(3) using Equation 3.5 .
Solution
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