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Mathematical Minimum (for Aspiring Physicists)

Derivation of some Differential Formulas

Differentiation of power function (for n 0, see Eq.(32) for the case with n = 0):

d(yn) (y + y)n - yn yn (1 + y/y)n - yn yn (1 + ny/y) - yn nyn-1y = nyn-1dy ---------- (26)

where we have used the power series expansion: (1 + x)n = 1 + nx + n(n-1)x2/2! + . In Eq.(10), we can re-arrange the y variable to one side and re-write y-2dy = -d(y-1) by applying the formula in Eq.(26) for the case of n = -1. Integration to obtain the solution becomes trivial once it is written in this form.

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.


Derviation of the differentiation for the sine and cosine functions:

dsinx = sin(x + x) - sinx = sinx cosx + cosx sinx - sinx cosx x = cosx dx ---------- (27)
dcosx = cos(x + x) - cosx = cosx cosx - sinx sinx - cosx -sinx x = -sinx dx ---------- (28)

where we have used the series expansion (derived from the Taylor series expansion about the point xo = 0):

sinx = x - x3/3! + ---------- (29a)
cosx = 1 - x2/2! + ----------(29b)

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.


§ Derivation of the relation d[ln(y)] = dy/y:

Logarithm Let us start from the differentiation of the logarithmic function loga(y) with an arbitrary base a:

d[loga(y)] loga(y + y) - loga(y) loga(1 + y/y)
(y/y) loga(1 + y/y)(y/y) ---------- (30)

where we have applied the logarithmic relations (Figure 11):
loga(u/v) = loga(u) - loga(v) ---------- (31a)
loga(u)n = n loga(u) ---------- (31b).

Figure 11 Logarithm
[view large image]
Now if we define n = y/y, e(1 + 1/n)n, a = e, ln(y) = loge(y), and use the identity loga(a) = 1; then
d[ln(y)] y/y = dy/y -------- (32).

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

d(ex) = ex dx ---------- (33).

Incidentally, logarithm is not equal to "log-a-rhythm" (Figure 12). It was invented for manipulating large number by reducing it to a manageable size, e.g., log10(1000000000) = log10(109) = 9. The first logarithmic table was constructed by the Scottish mathematician John Napier (1550-1617) in the base of e (since known as natural log). Also note that in term of complex function with a real and an imaginary parts, the exponential and trigonometric functions are related by the formula:

Log-a-Rhythm Euler's Formula y = ei 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 xo = 0 for ex, substituting to the independent variable with ix and comparing the result with Eqs.(29a,b). The Euler's divine formula (Figure 13) :

ei + 1 = 0 ---------- (35)

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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