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Section2_4 - Section 2.4 Derivatives of power functions and...

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(3/19/08) Section 2.4 Derivatives of power functions and linear combinations Overview: In this section we derive rules for differentiating power functions y = x n and linear combinations y = Af ( x ) + Bg ( x ) of functions whose derivatives are known. Topics: Leibniz notation and the differentiation operator The derivative of y = x n Derivatives of linear combinations of functions Leibniz notation and the differentiation operator Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716), who are considered to be the founders of calculus, each introduced notation for the derivative. Newton employed notation similar to the “prime notation” f prime ( a ) that we used in the last section. The symbols used by Leibniz evolved into what is known today as Leibniz notation . With this notation, the derivative f prime of f is denoted df dx , its value at x = a is denoted df dx vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle x = a , and the Δ x -formulation of the definition of the derivative from Section 2.3 becomes df dx vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle x = f prime ( x ) = lim Δ x 0 Δ f Δ x = lim Δ x 0 f ( x + Δ x ) - f ( x ) Δ x . (1) As we saw in the last section, Δ x is the run from ( x, f ( x )) to ( x + Δ x, f ( x + Δ x )) on the secant line, as shown in Figure 1, and Δ f is the corresponding rise f ( x + Δ x ) - f ( x ). x y y = f ( x ) x f ( x ) x + Δ x Δ f Δ x f ( x + Δ x ) Secant line of slope Δ f Δ x = f ( x + Δ x ) - f ( x ) Δ x FIGURE 1 One advantage of Leibniz notation is that it includes the name of the variable with respect to which the derivative is being taken. It is also used in the symbol d/dx for the differentiation operator that transforms a function f into its derivative df/dx . We write d dx [ f ( x )] = df dx vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle x or with prime notation for the derivative d dx [ f ( x )] = f prime ( x ) . 107
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p. 108 (3/19/08) Section 2.4, Derivatives of power functions and linear combinations The differentiation operator d/dx provides a convenient way to express derivatives of functions given by formulas. We will use it in the following discussion of the derivative d dx (1) of the function y = 1, the derivative d dx ( x ) of the function y = x , and other cases of d dx ( x n ) with constant n . The derivatives of y = 1 and y = x Because the constant function y = 1 and the function y = x are linear, their derivatives are the slopes of the lines that are their graphs (Figures 2 and 3): d dx (1) = 0 and d dx ( x ) = 1 . (2) We can also obtain these formulas from formula (2) for the derivatives, as in the following example. Example 1 Derive formulas (2) from definition (1) . Solution If f ( x ) = 1, then f ( x + Δ x ) = 1 for any Δ x and by (1) , d dx (1) = lim Δ x 0 1 - 1 Δ x = lim Δ x 0 (0) = 0 . For f ( x ) = x , we have f ( x + Δ x ) = x + Δ x d dx ( x ) = lim Δ x 0 ( x + Δ x ) - x Δ x = lim Δ x 0 (1) = 1 .
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