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Unformatted text preview: Essential techniques of Calculus J. R. Chasnov Department of Mathematics Hong Kong University of Science and Technology Fall 2007 1 Differentiation 1.1 Definition of derivative The derivative of the function y = f ( x ), denoted as f ′ ( x ) or dy/dx , is defined as the slope of the tangent line to the curve y = f ( x ) at the point ( x, y ). This slope is obtained by a limit. First, a line is drawn which passes through the points ( x, f ( x )) and ( x + h, f ( x + h )) (see Fig. 1). The slope of this line is the rise: f ( x + h ) − f ( x ) divided by the run: ( x + h ) − x = h . As h → 0, the drawn line approaches the tangent line. The derivative is thus defined as the limit f ′ ( x ) = lim h → f ( x + h ) − f ( x ) h . (1) Note that taking the limit is not the same as directly substituting h = 0, which gives f ′ ( x ) = 0 / 0, an undefined value that is not of much use. Commonly, we will need to cancel the h in the denominator against a factor of h in the numerator before setting h = 0 and evaluating the derivative. x x+h f(x) f(x+h) y=f(x) slope = [f(x+h)f(x)]/h Figure 1: The derivative as the limit of a slope. 1 The notation dy/dx for the derivative of y = f ( x ) comes from denoting the change h in the dependent variable x by Δ x , pronounced “delta x,” and the resulting change in the dependent variable y by Δ y , where Δ y = f ( x + Δ x ) − f ( x ) . Therefore, (1) can be rewritten in this new notation as f ′ ( x ) = lim Δ x → Δ y Δ x = dy dx , the second line defining dy/dx = f ′ ( x ). Note that strictly speaking dy/dx is defined after taking the limit and is a single entity. Sometimes, however, it is expedient to treat dy/dx as a fraction. Next we derive some general rules that will be useful in evaluating the deriva tives of complicated functions. Then we will determine specific rules differentiat ing some simple functions that can be used to construct complicated functions. 1.2 Derivative of the sum or difference of two functions Let h ( x ) = f ( x ) + g ( x ). We have h ′ ( x ) = lim Δ x → h ( x + Δ x ) − h ( x ) Δ x = lim Δ x → f ( x + Δ x ) + g ( x + Δ x ) − ( f ( x ) + g ( x )) Δ x = lim Δ x → f ( x + Δ x ) − f ( x ) Δ x + lim Δ x → g ( x + Δ x ) − g ( x ) Δ x = f ′ ( x ) + g ′ ( x ) . The result can be written as ( f + g ) ′ = f ′ + g ′ . In a similar fashion, ( f − g ) ′ = f ′ − g ′ . These simple rules generalize to the sum or difference of an arbitrary number of functions. 1.3 Derivative of the product of two functions If h ( x ) = f ( x ) g ( x ), then h ′ ( x ) = lim Δ x → h ( x + Δ x ) − h ( x ) Δ x 2 = lim Δ x → f ( x + Δ x ) g ( x + Δ x ) − f ( x ) g ( x ) Δ x = lim Δ x → f ( x + Δ x ) g ( x + Δ x ) − f ( x ) g ( x + Δ x ) + f ( x ) g ( x + Δ x ) − f ( x ) g ( x ) Δ x = lim Δ x → f ( x + Δ x ) − f ( x ) Δ x g ( x + Δ x ) + lim Δ x → f ( x ) g ( x + Δ x ) − g ( x ) Δ x = f ′ ( x ) g ( x ) +...
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 Spring '09
 T.Qian
 Calculus, Derivative

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