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**Unformatted text preview: **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 ) = 110sin(120 x ). This is the composite of f ( x ) = 110sin x and g ( x ) = 120 x . 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 ))....

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