02Techniques of Differentiation - 3 = + y x ? Sum and...

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T ECHNIQUES OF D IFFERENTIATION – S UM AND D IFFERENCE Fortunately, we don’t always have to compute derivates from the definition. There are some rules that make differentiation much easier. Constant Rule If f is a constant function, c x f = ) ( , then , or 0 ) ( = c dx d . 0 ) ( ) ( lim 0 = - = - + h c c h x f h x f h (Show graphically) Power Rule If n x x f = ) ( , then , for , I n 0 n , or 1 - = n n nx x dx d . a x a xa a x x a x a x a x a x a f x f a f n n n n a x n n a x a x - + + + + - = - - = - - = - - - - ) ...... )( ( lim lim ) ( ) ( lim ) ( 1 2 2 1 1 1 ) 2 ( 2 1 ..... lim - - - - - = + + + = n n n n n a x nx a xa a x x Ex . Find the derivative. 100 x y = Practice . Find the equation of the tangent line to the curve 6 x y = at 2 - = x . Ex . Differentiate 3 1 ) ( x x f = and x y = . Constant Multiple Rule 1 ) ( - = n nx x f 0 ) ( = x f ) ( ) ( x f c x g =
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If ) ( ) ( x cf x g = , then , or [ ] ) ( ) ( x f dx d c x cf dx d = . Ex . Differentiate 3 8 ) ( x x f = , 3 8 6 x y = . Ex . At what points on the hyperbola 12 = xy is the tangent line parallel to the line
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Unformatted text preview: 3 = + y x ? Sum and Difference Rules Sum Rule Suppose ) ( x f ′ and ) ( x g ′ exist. Then if ) ( ) ( ) ( x g x f x G + = , then ) ( x G ′ exists and ) ( ) ( ) ( x g x f x G ′ + ′ = ′ or . Difference Rule [ ] ) ( ) ( ) ( ) ( x g dx d x f dx d x g x f dx d-=-[ ] ) ( ) ( ) ( ) ( x g dx d x f dx d x g x f dx d + = + Ex . If 1 ) ( 2 3 4 +-+-= x x x x x f , find the equation of the tangent to the graph of f at the point (1, 1). Ex . Find the equation of the tangent line to x x y 3 7-= at (1, 4). Assign: p.198#1-3, 5-9, 11-14, 33-35, Worksheet on Derivatives...
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This note was uploaded on 07/12/2011 for the course MATH 241 taught by Professor Kim during the Fall '08 term at University of Illinois, Urbana Champaign.

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02Techniques of Differentiation - 3 = + y x ? Sum and...

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