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Unformatted text preview: Chapter 4 The Derivative In our investigation so far, we have defined the notion of an instantaneous rate of change, and called this the derivative. We have also identified this mathematical concept with the slope of a tangent line to the graph of a function. Recall that our definition for the derivative of a function, y = f ( x ) is as follows: Definition: The derivative of a function y = f ( x ) is dy dx = f prime ( x ) = lim h f ( x + h )- f ( x ) h In this chapter, we will use this definition to calculate the derivatives of power functions. In our previous discussions, we observed that power functions are building blocks of polynomials, a family of well-behaved functions that are exceedingly useful in approximations. Using some further elementary properties of derivatives we will arrive at a simple way of calculating the derivative of any polynomial. This will permit interesting and useful calculations, on a variety of applied problems. In this and the following sections, we will gain experience with the many-faceted properties of derivatives that use relatively simple differentiation calculations. (Some of the problems we address will be challenging nevertheless, but all of them will be based on polynomial and power function forms.) In our examples below we use the definition of the derivative, to compute derivatives of a few simple power functions. 4.1 Computing the derivative Example: Compute the derivative of the first power functions y = f ( x ) = x n where n = 0 , 1 , 2 , 3 , . . . v.2005.1 - September 23, 2009 1 Math 102 Notes Chapter 4 For y = f ( x ) = Cx 2 we have dy dx = lim h f ( x + h )- f ( x ) h = lim h C ( x + h ) 2- Cx 2 h = lim h C ( x 2 + 2 xh + h 2 )- x 2 h = lim h C (2 xh + h 2 ) h = lim h C (2 x + h ) = C (2 x ) = 2 Cx (4.1) For y = f ( x ) = Kx 3 we have dy dx = lim h f ( x + h )- f ( x ) h = lim h K ( x + h ) 3- Kx 3 h = lim h K ( x 3 + 3 x 2 h + 3 xh 2 + h 3 )- x 3 h = lim h K (3 x 2 h + 3 xh 2 + h 3 ) h = lim h K (3 x 2 + 3 xh + h 2 ) = K (3 x 2 ) = 3 Kx 2 (4.2) For the simpler functions, y = C and y = Bx , we could use a simple argument to determine the derivative: Both these functions represent straight lines (of slopes 0 and B, respectively). Since these slopes are the same everywhere, the derivatives of these functions are 0 and B respectively. (The reader can also establish these facts with calculations similar to the above.) With these calculations we find the following pattern of derivatives for the power functions. (In the table below, we have taken all the coefficients to be 1 in previous examples, for simplicity of the presentation.) The power rule of differentiation Function Derivative f ( x ) f prime ( x ) 1 x 1 x 2 2 x x 3 3 x 2 ....
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