Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/49

 

CHAPTER IV

 

DIFFERENTIATION

 

25. Introduction. We shall now proceed to investigate the manner in which a function changes in value as the independent variable changes. The fundamental problem of the Differential Calculus is to establish a measure of this change in the function with mathematical precision. It was while investigating problems of this sort, dealing with continuously varying quantities, that Newton^[1] was led to the discovery of the fundamental principles of the Calculus, the most scientific and powerful tool of the modern mathematician.

26. Increments. The increment of a variable in changing from one numerical value to another is the difference found by subtracting the first value from the second. An increment of $\scriptstyle {x}$ is denoted by the symbol $\scriptstyle {\Delta x}$, read delta $\scriptstyle {x}$.

The student is warned against reading this symbol delta times $\scriptstyle {x}$, it having no such meaning. Evidently this increment may be either positive or negative^[2] according as the variable in changing is increasing or decreasing in value. Similarly,

$\scriptstyle {\Delta y}$	denotes an increment of $\scriptstyle {y}$,
$\scriptstyle {\Delta \phi }$	denotes an increment of $\scriptstyle {\phi }$,
$\scriptstyle {\Delta f(x)}$	denotes an increment $\scriptstyle {f(x)}$, etc.

If in $\scriptstyle {y=f(x)}$ the independent variable $\scriptstyle {x}$, takes on an increment $\scriptstyle {\Delta x}$, then $\scriptstyle {\Delta y}$ is always understood to denote the corresponding increment of the function $\scriptstyle {f(x)}$ (or dependent variable $\scriptstyle {y}$).

The increment $\scriptstyle {\Delta y}$ is always assumed to be reckoned from a definite initial value of $\scriptstyle {y}$ corresponding to the arbitrarily fixed initial value of $\scriptstyle {x}$ from which the increment $\scriptstyle {\Delta x}$ is reckoned. For instance, consider the function$\scriptstyle {y=x^{2}}$.

1. Sir Isaac Newton (1642–1727), an Englishman, was a man of the most extraordinary genius. He developed the science of the Calculus under the name of Fluxions. Although Newton had discovered and made use of the new science as early as 1670, his first published work in which it occurs is dated 1687, having the title Philosophiae Naturalis Principia Mathematica. This was Newton's principal work. Laplace said of it, "It will always remain preëminent above all other productions of the human mind." See frontispiece.
2. Some writers call a negative increment a decrement.