Stewart6e3_1

# Stewart6e3_1 - MATH 191 Sections 1 and 3 Calculus I Fall...

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MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 3.1: At Last! Shortcuts! Recall the definition (and notation) for the derivative of a function f : if y = f ( x ), we can write f ( x ) or dy dx for the derivative. Our goal right now is to start developing some shortcut formulas that will allow us to do away with the clumsy limit formulas that become a pain in the rear when f is even mildly complicated. Let’s start off with a simple verification, and show that for any constant c , d dx ( c ) = 0. Not bad, but we can do better. Let’s prove another easy one: d dx ( x ) = 1:

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Thus we now know that d dx x 1 = 1 x 0 . A few classes back, we showed that d dx x 2 = 2 x . Do you see a pattern yet? Let’s do one more: what’s d dx x 3 ? Now we’re talkin’! Let’s go out on a limb and guess The Power Rule. If n is a non-negative integer, then d dx x n = . Proof. It’s easier to find the derivative at a given point a : f ( a ) = lim x a x n - a n x - a . Once we’ve written out this formula, we factor out x - a from the top and simplify:
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