16 DERIVATIVE.docx - DERIVATIVE The derivative of f at x denoted by f x)= f(x for dy dx is defined as follows dy =� dx Derivatives of a Constant f(x =

# 16 DERIVATIVE.docx - DERIVATIVE The derivative of f at x...

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DERIVATIVE The derivative of f at x, denoted by f ' ( x ) for dy dx is defined as follows f ' ( x ) = dy dx = ¿ Derivatives of a Constant f(x) = c (constant) then f ’(x) = 0 Eamples: 1. y ¿ 6 dy dx ¿ 0 2. y ¿ 4 5 dy dx ¿ 0 Basic Power Rule dx d ( x n ) = nx n 1 Eamples: 1. y ¿ x 6 dy dx ¿ 6 x 5 2. y ¿ x 4 3 + x 3 dy dx ¿ 4 3 x 1 3 3 x 4 3. y ¿ x 2 5 + 1 x 2 dy dx ¿ 2 5 x 3 5 2 x 3 Basic Chain Rule Eamples: 1. y ¿ ( 3 x + 2 ) 4 dy dx ¿ 4 ( 3 x + 2 ) 3 3 2. y ¿ ( 6 x 2 4 x + x ) 6 dy dx = 6 ( 6 x 2 4 x + x ) 5 ( 12 x 4 + 1 2 x )

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Product Rule dx d { f ( x ) g ( x ) } = f ' ( x ) g ( x )+ f ( x ) g ' ( x ) Eamples: 1. y ¿ ( 2 x + 3 ) x 3 dy dx ¿ 3 x 2 ( 2 x + 3 )+ 2 x 3 2. y ¿ ( x 2 + 4 )( 3 x + 5 ) dy dx ¿ 2 x ( 3 x + 5 )+ 3 ( x 2 + 4 ) Chain Rule (extended) 1. y ¿ ( 4 x 2 6 x )( x 2 3 ) 5 dy dx ¿ ( 4 x 2 6 x ) 5 ( x 2 3 ) 4 2 x +( 8 x 6 )( x 2 3 ) 5 Quotient Rule dx d { f ( x ) g ( x ) } = f ' ( x ) g ( x )− f ( x ) g ' ( x ) [ g ( x ) ] 2 Eamples: 1. y ¿ x 2 + 3 x 4 dy dx = ( 2 x ) x 4 −( x 2 + 3 )( 4 x 3 ) [ x 4 ] 2 2. y ¿ ( x 2 + 4 )( 3 x + 5 ) dy dx ¿ 2 x ( 3 x + 5 )+( x 2 + 4 ) 3 Derivative of trigonometric functions f ( x ) f ' ( x ) a) sinx cosx b) cos x sin x

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