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Unformatted text preview: MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 3.4: The Chain Rule! The for derivatives tells us how to differentiate of functions. It is by far the most important rule for differentiation. Need proof? Have you noticed the contrived simplicity of almost every func- tion weve considered so far? Almost any reasonably complicated function is inexplicable without the use of composition in some way! Examples. With the rules weve studied to date, we still dont know how to differentiate the functions x + 1 and sin( x- 1), as simple as these are. In the space below, come up with at least two more functions whose derivatives still elude us: Okay, to approach the rule well call the Chain Rule (for chaining one function after another), lets try to get some idea as to what the derivative a composition ( f g )( x ) = f ( g ( x )) should be. Suppose that u = g ( x ) and y = f ( u ), so that y = f ( u ) = f ( g ( x )). Suppose also that 1. at the point x = a , g (...
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