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chain rule notes

# chain rule notes - The Chain Rule The chain rule is the...

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The Chain Rule The chain rule is the rule for finding the derivative of composite functions. A composite function is one formed in the following manner. We say F is the composite of f and g , written F = f g , if F ( x ) = f ( g ( x )). So in the formula for f ( x ), we substitute for x the formula for g ( x ). Examples. If f ( x ) = x 2 +3 x +4 and g ( x ) = 5 x +6 then f ( g ( x )) = (5 x +6) 2 +3(5 x +6)+4 = 25 x 2 + 75 x + 58 . Compare g ( f ( x )) = 5( x 2 + 3 x + 4) + 6 = 5 x 2 + 15 x + 6. Let f ( x ) = x and g ( x ) = 25 - x 2 . Then F ( x ) = f ( g ( x )) = p 25 - x 2 and g ( f ( x )) = 25 - ( x ) 2 = 25 - x (but note, the domain of g ( f ( x )) is x 0). Let F ( x ) = 110 sin(120 πx ). This is the composite of f ( x ) = 110 sin x and g ( x ) = 120 πx . The function F ( x ) = 1 + x 1 - x 5 is the composite of f ( x ) = x 5 and g ( x ) = 1 + x 1 - x . Here is another way of writing compositions: If y = f ( u ) and u = g ( x ) then y = f ( g ( x )). For example, we may express F ( x ) = 1 + x 1 - x 5 by writing y = u 5 and u = 1 + x 1 - x . The chain rule. Suppose F ( x ) = f ( g ( x )). Then F 0 ( x ) = f 0 ( g ( x )) g 0 ( x ). Examples. Let F ( x ) = ( x 2 + 1) 6 . Here, f ( x ) = x 6 and g ( x ) = x 2 + 1, so f 0 ( x ) = 6 x 5 and g 0 ( x ) = 2 x . Also, f 0 ( g ( x )) = 6( x 2 + 1) 5 . So F 0 ( x ) = f 0 ( g ( x )) g 0 ( x ) = 6( x 2 + 1) 5 (2 x ).

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chain rule notes - The Chain Rule The chain rule is the...

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