121616949-math.65 - 3.3 The Product Rule 51 the general...

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3.3 The Product Rule 51 the general calculation even without knowing the answer in advance. d dx ( f ( x ) g ( x )) = lim Δ x 0 f ( x + Δ x ) g ( x + Δ x ) - f ( x ) g ( x ) Δ x = lim Δ x 0 f ( x + Δ x ) g ( x + Δ x ) - f ( x + Δ x ) g ( x ) + f ( x + Δ x ) g ( x ) - f ( x ) g ( x ) Δ x = lim Δ x 0 f ( x + Δ x ) g ( x + Δ x ) - f ( x + Δ x ) g ( x ) Δ x + lim Δ x 0 f ( x + Δ x ) g ( x ) - f ( x ) g ( x ) Δ x = lim Δ x 0 f ( x + Δ x ) g ( x + Δ x ) - g ( x ) Δ x + lim Δ x 0 f ( x + Δ x ) - f ( x ) Δ x g ( x ) = f ( x ) g ( x ) + f ( x ) g ( x ) A couple of items here need discussion. First, we used a standard trick, “add and subtract the same thing”, to transform what we had into a more useful form. After some rewriting, we realize that we have two limits that produce f ( x ) and g ( x ). Of course, f ( x ) and g ( x ) must actually exist for this to make sense. We also replaced lim Δ x 0 f ( x x ) with f ( x )—why is this justified? What we really need to know here is that lim x a f ( x ) = f ( a ), that is, the values of f ( x ) get close to f ( a ) as x gets close to a . This is certainly not always true, as in the “greatest integer” or “floor” function, for example. The value of the function f ( x ) = x is x rounded down” or “the greatest integer less than or equal to x .” For example, 4 . 5 = 4, 47 = 47. The graph of this function is indicated in figure 3.1 . As x gets close to 1 from the left, f ( x ) gets close to 0 (it actually
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