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d(y

where we have used the power series expansion: (1 + x)

Note: Only the highest term or the 2nd highest term (in case of cancellation of the highest term) is taken in the series expansion. Readers of this page are strongly urged to complete the intermediate steps for many of the derivations as it offers a chance to apply their skill on the subjects.

dsinx = sin(x + x) - sinx = sinx cosx + cosx sinx - sinx

dcosx = cos(x + x) - cosx = cosx cosx - sinx sinx - cosx

where we have used the series expansion (derived from the Taylor series expansion about the point x

sinx = x - x

cosx = 1 - x

and the trigonometric formulas:

sin(A + B) = sinAcosB + cosAsinB ---------- (29c)

cos(A + B) = cosAcosB - sinAsinB ---------- (29d)

These trigonometric formulas were discovered more than one thousand years ago by the Muslim mathematician Abu Wafa.

Let us start from the differentiation of the logarithmic function log_{a}(y) with an arbitrary base a: d[log _{a}(y)] _{} log_{a}(y + y) - log_{a}(y) _{} log_{a}(1 + y/y) _{} (y/y) log _{a}(1 + y/y)^{(y/y)} ---------- (30)where we have applied the logarithmic relations (Figure 11): log _{a}(u/v) = log_{a}(u) - log_{a}(v) ---------- (31a)log _{a}(u)^{n} = n log_{a}(u) ---------- (31b).
| |

## Figure 11 Logarithm |

d[ln(y)]

As a corollary from the definition of the exponential base e, it can be shown that differentiation of the exponential function e

d(e

Incidentally,

y = e^{i x} = cos x + i sin x ---------- (34)This relationship was discovered by the Swiss mathematician Leonhard Euler (1707-83). The modern derivation can be obtained easily if we perform a Taylor series expansion about the point x _{o} = 0 for e^{x}, substituting to the independent variable with ix and comparing the result with Eqs.(29a,b). The Euler's divine formula (Figure 13) :e ---------- (35)
^{i} + 1 = 0 | ||

## Figure 12 Log-a-Rhythm [view large image] |
## Figure 13 Euler's Formula [view large image] |
is a straight forward application of Eq.(34). It contains all the special symbols in mathematics: e, i = , , 1, and 0. |

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