121616949-math.89 - 4.7 Derivatives of the exponential and...

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4.7 Derivatives of the exponential and logarithmic functions 75 As with the sine, we don’t know anything about derivatives that allows us to compute the derivatives of the exponential and logarithmic functions without going back to basics. Let’s do a little work with the definition again: d dx a x = lim Δ x 0 a x x - a x Δ x = lim Δ x 0 a x a Δ x - a x Δ x = lim Δ x 0 a x a Δ x - 1 Δ x = a x lim Δ x 0 a Δ x - 1 Δ x There are two interesting things to note here: As in the case of the sine function we are left with a limit that involves Δ x but not x , which means that whatever lim Δ x 0 ( a Δ x - 1) / Δ x is, we know that it is a number, that is, a constant. This means that a x has a remarkable property: its derivative is a constant times itself. We earlier remarked that the hardest limit we would compute was lim x 0 sin x/x = 1; we now have a limit that is just a bit too hard to include here. In fact the hard part is to see that lim Δ x 0 ( a Δ x - 1) / Δ x even exists—does this fraction really get closer and closer to some fixed value? Yes it does, but we will not prove this fact. We can look at some examples. Consider (2 x - 1) /x for some small values of
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