It is clear, then, the smaller the difference between

a and b, the more nearly does

a2-b2

a-b

approximate to

2a; and, therefore, we say that ultimately, in the

limit, when b=a,

a2-b2

=2a.

a-b

(See also Art. 91.)

V. Function-Differential Co-efficient—Differential Coefficient of a Simple Function.

21. If one quantity depend upon a particular value of another variable quantity, the first quantity is said to be a Function of the second; or, if one quantity or expression involve another in any form, it is said to be a Function of that quantity. The quantity upon which the other depends is called the independent variable, and the function the dependent variable.

rx

Thus 3x, x2, etc., are all functions of x. The independent variable is x, upon whose value the value of the expression, or function of x, depends; similarly the area of a square is a function of its side, the side being the independent variable, upon whose value the value of the area depends; the volume of a cube is a function of its edge; the circumference and area of a circle are, each of them, functions of its radius; the volume of a spere is a function of its radius-the edge of the cube, the radius of the circle, and the radius of the sphere being the independent variables in each case, and the volume of cube, area, and circumference of the circle, and the volume of the sphere, the dependent variables.

22. Our object is to find the ratio of the rate of variation (i.e., the rate of increase or decrease) of the function to the rate of variation of the independent variable, as the independent variable undergoes infini

tesimally small variations. This ratio is called the Differential Co-efficient of the function.

23. If a variable quantity increase uniformly, the function either increases uniformly or accordingly to any variable law.

Let x be a variable quantity, and let it increase uniformly by the quantities 1, 1, 1, etc.

Then the successive values will be

x+1, x+2, x+3, etc.

Then also any number of times of x will increase uniformly-say 3x-the values being

3x+3, 3x+6, 3x+9, etc.,

which increase uniformly by 3.

Again, take px, then the successive values are px+p, px+2p, px+3p, etc.,

which increase uniformly by p.

Further, let x be a variable quantity, and let it increase uniformly by the quantities a, a, a, etc., then it will, at the successive stages, become

x+a, x+2α, x+3a, etc.,

and, as before, any number of times of x will increase uniformly.

First, take 3x, then the successive values become 3x+3a, 3x+6a, 3x+9a, etc.,

which increase uniformly by 3a.

Next, take px, then the successive values become px+pa, px+2pa, px+3pa, etc.

which increase uniformly by pa.

24. It is evident that if a constant quantity (i.e., one which does not vary) be connected with the function px by the sign + or -, the function will still increase uniformly, for the successive values will be

px+pa+C, px+2pa+C, px+3pa+C, etc., 25. Again, to illustrate this geometrically, suppose we have a straight line AB, and draw AC, making any acute angle with AB, and let a variable straight line Pp move from A so as to remain always perpendicular to AB, and have one extremity in AB and the other in

AC, and take up, at successive periods, such positions as P'p', P"p", etc.; then it is evident that, as Ap in

creases uniformly, and becomes Ap', Ap", etc., Bp increases uniformly, for

and so for any other position of Pp.

Again, let DE be parallel to AB, and let Pq be the new variable line.

Now Pq=Pp+pq, and pq is constant; and it is

evident that, as Ap increases uniformly, Pq increases at the same rate as before.

26. Now let x be any given variable quantity, and 3x a given function of x, then as x becomes x+h, 3x becomes 3(x+h) or 3x + 3h, and the ratio of the rate of increase in the function to the rate of increase in 3h 3 = 3. 1

the variable=

h

Now let h become less and less; this ratio still holds good, and, ultimately, when h is indefinitely diminished, i.e., in the limit, the rate of (increase in this case) variation in the function is to the rate of variation of the independent variable as 3: 1, i.e., the differential co-efficient of 3x is 3, and a similar argument will hold if we take nx instead of 3x. Thus it will be found generally that

the differential co-efficient of nx with

respect to x, i.e., where x is the inde- =n.
pendent variable,

27. Let us take a quantity +C, where x is the independent variable, and take n(x+C) as a function of this, C being constant; and let x receive a small increment and become x + h

then x+C becomes (x+h)+C,

and n(x+C) becomes n(x + h)+nC,

or nx+nh +nC,

and the ratio of the rate of variation of the function

to the rate of variation of the variable=

nh
=2.
h

NOTE. It is obvious that if the rate at which two quantities increase be added together, the sum will be the rate of increase at which the sum of the quantities increases; and the difference, the rate at which the difference increases. Therefore, if we have two functions of the same variable connected by the signs + or –, the differential co-efficient of the whole expression will be the sum or difference of the differential co-efficients of the two parts.

VI. Differential Co-efficient of x2.

28. Let a square have a side of 4 feet, then the area of the square=16 square feet, that is if

x=4 x2=16.

Now, suppose the side to receive a small increment and become 4001 feet, then the square becomes 16.008001 square feet.

If we omit 000001, then the ratio of the increase of the function to the increase of the variable, or of

008: 0018:1

=twice side of square: 1.

Again, suppose the side to receive a still smaller increment and become 4.000001 feet; then the area of the square=16*000008000001 square feet.

Here by omitting 000000000001 we commit an almost inappreciable error, and, as before, and still more truly, the ratio of the increase of the function to the increase of the variable is

*000008: 000001, or 8: 1,

or 2 x side of square : 1.

Therefore we may state that, ultimately, when the increment of the side is indefinitely diminished, or in other words is made indefinitely small, the ratio of the rate of increase in the function (square) to the rate of increase of the variable (side) is 2x: 1,

or the differential co-efficient of x2 is 2x.

29. Let AB be a straight line, and let a square be described on AB. Then this square is a function of AB. Now let AB receive a small increment BC; the straight line has now become AC, and the square has, in consequence, received an increment of the two shaded rectangles and the small square a.

Let the straight line receive a further increment CD (BC), then the square will have received an increment of four rectangles and four such squares as a.

Now let the straight line receive a further increment DE (=CD=BC), then the square will have received an increment of six rectangles, such as the shaded rectangles, and nine such squares as a.

Thus we see that, as the straight line increases uniformly, the square increases, but not uniformly.

30. Again, when the side has an increment BC, the