05Techniques of DifferentiationIII

# 05Techniques of DifferentiationIII - x F ′ if 3 2 2 = x x...

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T ECHNIQUES OF D IFFERENTIATION – T HE C HAIN R ULE There is another technique to differentiate the composition of two functions. This is called the Chain Rule . If ) ( x g and )) ( ( x g f both exist, and g f F = is the composite function defined by )), ( ( ) ( x g f x F = then ) ( x F exists and is given by ) ( )) ( ( ) ( x g x g f x F = , or In Leibniz notation, if ) ( x f y = and ) ( x g y = are both differentiable functions, then dx dy dy dy dx dy = . Ex . Differentiate 1 ) ( 2 + = x x F . Ex . If 1 2 3 + + = u u y , where 1 2 2 - = x u , find 2 = x dx dy “the derivative at x = 2”. Ex . Find )

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Unformatted text preview: ( x F ′ if 3 2 ) ( 2 + = x x F . Special Case: Power Rule combined with Chain Rule If n is any real number and ) ( x g u = is differentiable, then dx dy nu u dx d n n 1 ) (-= or, [ ] [ ] ) ( ) ( ) ( 1 x g x g n x g dx d n n ′ =-. Ex . If 8 2 ) 2 ( +-= x x y , find dx dy . Ex . Differentiate 100 3 ) 1 (-= x y . Assign: p.209 #7-13, 35-39, 49, 50, Worksheet – no implicit differentiation...
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05Techniques of DifferentiationIII - x F ′ if 3 2 2 = x x...

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