SECTION 14.4 The Tangent Problem; The Derivative 963 Finding the Derivative of a Function at a Number Find the derivative of ( ) = − f x x x 2 5 2 at 2. That is, find ( ) ′ f 2 . Solution EXAMPLE 2 Since ( ) = ⋅ − ⋅ = − f 2 2 2 5 2 2, 2 we have ( ) ( ) ( ) ( ) ( )( ) − − = − − − − = − + − = − − − f x f x x x x x x x x x x 2 2 2 5 2 2 2 5 2 2 2 1 2 2 2 2 The derivative of f at 2 is ( ) ( ) ( ) ( )( ) ′ = − − = − − − = → → f f x f x x x x 2 lim 2 2 lim 2 1 2 2 3 x x 2 2 Now Work PROBLEM 21 Example 2 provides a way of finding the derivative at 2 analytically. Graphing utilities have built-in procedures to approximate the derivative of a function at a number. Consult the user’s manual for the appropriate keystrokes. Finding the Derivative of a Function at a Number Using a Graphing Utility Use a graphing utility to find the derivative of ( ) = − f x x x 2 5 2 at 2. That is, find ( ) ′ f 2 . EXAMPLE 3 Solution Figure 19 shows the solution using a TI-84 Plus CE graphing calculator.* As shown, ( ) ′ = f 2 3. Finding the Derivative of a Function at a Number Find the derivative of ( ) = f x x2 at c . That is, find ( ) ′ f c . Solution EXAMPLE 4 Since ( ) = f c c ,2 we have ( ) ( ) ( )( ) − − = − − = + − − f x f c x c x c x c x c x c x c 2 2 The derivative of f at c is ( ) ( ) ( ) ( )( ) ′ = − − = + − − = → → f c f x f c x c x c x c x c c lim lim 2 x c x c As Example 4 illustrates, the derivative of ( ) = f x x2 exists and equals c2 for any number c . In other words, the derivative is itself a function, and using x for the independent variable, we can write ( ) ′ = f x x2 . The function ′ f is called the derivative function of f or the derivative of f . We also say that f is differentiable . The instruction “differentiate f ” means “find the derivative of f. ” Now Work PROBLEM 33 3 Find Instantaneous Rates of Change The average rate of change of a function f from c to x is Figure 19 ( ) ( ) = Δ Δ = − − ≠ y x f x f c x c x c Average rate of change *The TI-84 Plus CE uses an alternative notation for the derivative of f at c , namely ( ) = d dx f x | . x c Need to Review? Average rate of change is discussed in Section 2.3, pp. 93 – 94 .
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