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Unformatted text preview: Brief Review of Calculus The Derivative Definition The derivative of the function f is the function f defined by: f ( x ) = lim h f ( x + h ) f ( x ) h , for all x for which the limit exists. If y = f ( x ), we often write: dy dx = Dy = f ( x ) . Derivative of a Constant If f ( x ) = c (a constant) for all x , then f ( x ) = 0 for all x . That is: dc dx = Dc = 0 . Example D (7) = 0 Power Rule for a Positive Integer If n is a positive integer and f ( x ) = x n , then f ( x ) = nx n 1 . Example D ( x 7 ) = 7 x 6 Derivative of a Linear Combination If f and g are differentiable functions and a and b are fixed real numbers, then: D [ af ( x ) + bg ( x )] = aDf ( x ) + bDg ( x ) . With u = f ( x ) and v = g ( x ), this takes the form: d ( au + bv ) dx = a du dx + b dv dx . Example D (16 x 6 ) = 16 6 x 5 = 96 x 5 Example D (36 + 26 x + 7 x 5 5 x 9 ) = 0 + 26 1 + 7 5 x 4 5 9 x 8 = 26 + 35 x 4 45 x 8 The Product Rule If f and g are differentiable at x , then fg is differentiable at x , and: D [ f ( x ) g ( x )] = f ( x ) g ( x ) + f ( x ) g ( x ). With u = f ( x ) and v = g ( x ), this product rule takes the form: d ( uv ) dx = u dv dx + v du dx . When it is clear what the independent variable is, we can make the product rule even briefer: ( uv ) = u v + uv . 1 Example If f ( x ) = (1 4 x 3 )(3 x 2 5 x + 2), then: f ( x ) = [ D (1 4 x 3 )](3 x 2 5 x + 2) + (1 4 x 3 )[ D (3 x 2 5 x + 2)] = ( 12 x 2 )(3 x 2 5 x + 2) + (1 4 x 3 )(6 x 5) = 60 x 4 + 80 x 3 24 x 2 + 6 x 5 The Reciprocal Rule If f is differentiable at x and f ( x ) 6 = 0, then: D 1 f ( x ) = f ( x ) [ f ( x )] 2 . With u = f ( x ), the reciprocal rule takes the form: d dx 1 u = 1 u 2 du dx . If there can be no doubt what the independent variable is, we can write: 1 u = u u 2 . Example D 1 x 2 + 1 = D ( x 2 + 1) ( x 2 + 1) 2 = 2 x ( x 2 + 1) 2 Power Rule for a Negative Integer If n is a negative integer, then Dx n = nx n 1 ....
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This note was uploaded on 01/08/2012 for the course EXST 4050 taught by Professor Staff during the Fall '10 term at LSU.
 Fall '10
 Staff

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