Page 106 - Calculus for the Clueless, Calc II

Not too shabby. More to come.

Example 43—

The series for x/(1 + x) :
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Differentiating, we get:

Multiply by -x; our result is:

When mathematicians do things like this, you tend to believe that mathematics can do everything and any thing.
However, this is not true. However, the best is yet to come!!!!!!

We can derive every property of the sine and cosine using infinite series, never, never mentioning triangles or
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angles. Amazing, huh?! Given sin x = x - x /3! + x /5! ... and cos x = 1 - x /2! + x /4! - x /6!.... How about cos
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2x? cos 2x = 1 - 2x + (2/3)x -(4/45)x ... [2x for x in cos x].
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How about sin x + cos x = 1?
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2

How about tan x? Well we would like tan x = sin x/cos x.

How about derivatives? If f(x) = sin x, we want f'(x) = cos x.