Illustration of the Rules for Differentiation

Excerpts

• Illustration of the Rules for Differentiation

  At the risk of being tedious to some readers, we will proceed to illustrate these formulæ by examples from the tables of logarithms and sines

• Illustration of the Rules for Differentiation

  Let $x = 1000$, whence $y = \text{common log}~ 1000 =3$; and let $dx = 1$, or let it be required to find the common logarithm of $1000 + 1$, or $1001$.

• Illustration of the Rules for Differentiation

  The tables give $3.0004341$, differing from the former only in the $7$thth place of decimals.

• Illustration of the Rules for Differentiation

  Let $x = 16°$, in which case $\sin x = .2756374$, and $\cos x = .9612617$.

• Illustration of the Rules for Differentiation

  These examples may serve to show how nearly the real ratio of two increments approaches to their limit, when the increments themselves are small.

Equations

• Illustration of the Rules for Differentiation

  y = \text{common log}~x

  The variable y is defined as the common logarithm of x.

• Illustration of the Rules for Differentiation

  .4342944 \left(\frac{dx}{x} - \tfrac{1}{2}\, \frac{(dx)^{2}}{x^{2}} + \tfrac{1}{3}\, \frac{(dx)^{3}}{x^{3}} - \etc.\right)

  The real increment of the common logarithm y when x becomes x + dx is given by an infinite series in dx/x, whose first term is .4342944 dx/x.

• Illustration of the Rules for Differentiation

  y = \sin x

  The variable y is defined as the sine of x.

• Illustration of the Rules for Differentiation

  \cos x\, dx - \frac{1}{2}\sin x\, (dx)^{2} - \etc.

  When x is increased by dx, sin x is increased by a series whose first term is cos x dx, of which only the first term is taken in the example.

Problems

No exercises in this chapter.