3_4 - Use the Chain Rule on Power function F ( x ) = [ g (...

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Section 3.4: The Chain Rule Instructor: Ms. Hoa Nguyen ([email protected]) Composite function y = F ( x ) is a composite function of the form y = ( f g )( x ) = f ( g ( x )). Examples : Find f and g to express the following y = F ( x ) as F ( x ) = ( f g )( x ) = f ( g ( x )). y = x 2 + 1 y = sin( x 2 ) y = sin 2 x y = ( x 3 - 1) 100 y = 1 3 x 2 + x +1 y = ( t - 2 2 t +1 ) 9 y = (2 x + 1) 5 ( x 3 - x + 1) 4 y = e sin x The Chain Rule If g is differentiable at x and f is differentiable at g ( x ), then the composite function F = f g defined by F ( x ) = f ( g ( x )) is differentiable at x and F 0 is given by the product F 0 ( x ) = d dx f ( g ( x )) = f 0 ( g ( x )) · g 0 ( x ) In other words, to perform the Chain Rule on a composite function F ( x ) = f ( g ( x )) , we take the derivative of the OUTER function f [at g ( x ) ], and then multiply with the derivative of the INNER function g [at x ] . In Leibniz notation , if y = f ( u ) and u = g ( x ) (i.e., y = ( f g )( x ) = f ( g ( x ))) are differentiable, then dy dx = dy du · du dx Remarks : dy dx = y 0 with respect to x while dy du = y 0 with respect to u . These are Leibniz notations , NOT a quotient. Example 1 of Section 3.4 (textbook): Example 2 of Section 3.4 (textbook):
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Unformatted text preview: Use the Chain Rule on Power function F ( x ) = [ g ( x )] n ⇒ F ( x ) = n [ g ( x )] n-1 · g ( x ) where n is any real number and g ( x ) exists (or g is differentiable). Example 3 of Section 3.4 (textbook): Example 4 of Section 3.4 (textbook): Example 5 of Section 3.4 (textbook): Example 6 of Section 3.4 (textbook): Use the Chain Rule on Exponential function F ( x ) = e g ( x ) ⇒ F ( x ) = e g ( x ) · g ( x ). What is the derivative of F ( x ) = a x , a > , a 6 = 1 ? F ( x ) = a x = ( e ln a ) x = e (ln a ) x ⇒ F ( x ) = a x ln a . Remarks : In Section 3.1, we learned that if f ( x ) = a x ( a > 0) then f ( x ) = f (0) a x . So, f (0) = ln a . Example 7 of Section 3.4 (textbook): Example 9 of Section 3.4 (textbook):...
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This note was uploaded on 05/23/2011 for the course MAC 2311 taught by Professor Noohi during the Fall '08 term at FSU.

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