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a being a positive quantity, it is plain that y will be imaginary whenever x is of the form

28 + 1

2u 1 and u being integers. Now between any two values of x, however little they may differ from each other, we may intercalate an infinite number of fractions of the above form. Thus we see that it is impossible to join by a continuous curve two points of the locus of the equation, corresponding to two systems of real values of x and y, however near they may be to each other. These anomalous functions are inapplicable to questions of natural philosophy, and have attracted but little attention even in pure analysis. In this treatise we shall direct our attention entirely to continuous functions.

8. Functions are distinguished also by the names of algebraical and transcendental. If y be connected with a by an

x equation involving only the ordinary operations of addition and subtraction, multiplication and division, evolution and involution of assigned degrees, y is said to be an algebraical function of x. If the equation of connection does not satisfy this condition, involving for instance exponential, logarithmic, or circular functions of x and y, then y is said to be a transcendental function of x. Thus, for examples, in the equations ya

y à 72 y is an algebraical function of x; and a transcendental one in the equations

a* + 2y = ctty, log (xy) = x + y,
x sin y + y sin x = sin (xy).

X = 9. It frequently happens that, if

F(x, y) = 0 be the equation to a curve, y will for a certain value of x experience a dissolution of continuity by becoming impossible, although the curve is itself continuous at the point. Thus if


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y = +, y will continuously vary from one value to another as a decreases from any assigned positive value down to zero, but, the moment z becomes negative, y becomes impossible. The two branches corresponding to the double sign, each of hich terminates abruptly at the origin, join together at this point and thus form a continuous curve.

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Definition of a Differential Coefficient. 10. LET y be a certain function of x, and let y' be the value assumed by y when x becomes x'. Then, as a keeps continuously approaching to the value of x, the fraction

y Y

x will continuously tend towards a certain value from which it will ultimately differ by a quantity less than any assignable magnitude, or, in other words, to which it will be ultimately equal. The indefinitely small values of the differences a' X, y' y, are usually denoted by the symbols dx, dy, and the ultimate value of the fraction

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In this expression du and dy are any quantities whatever, either finite or infinitesimal, which are in the ratio of the

dy ultimate values of 8x and dy. The fraction is called the

dx differential coefficient of y with regard to 2, the quantities dx and dy being called the differentials of x and y. The object of the Differential Calculus is to investigate the properties of differentials and differential coefficients, and to develop the general principles of their application to the theory of coordinate geometry and other branches of pure mathematics, and to the estimation of the phenomena of nature. Ex. Let y = x*: then y' = x": whence y' - y = 2" – x3 = (x' – x) (z"? + x'x + xo),


x2 + x'x + x2.

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When a approaches indefinitely near to x, the left-hand

dy member of the equation becomes and the right-hand

dx member assumes its limiting value 3x?: thus

dy – 3t",





dy = 3x dx, that is, the differential coefficient of ac with respect to x is 3x, and its differential is 3.cdc.

Differentiation of a Constant. 11. If y = C, where c denotes any constant quantity, that is,

y any quantity which does not experience variation in consequence of a variation in the value of x, then


dac For we have

y = 0, y = 0,

= 0.

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and therefore, proceeding to the limit,



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dx or the differential coefficient of a constant quantity is always



Differentiation of the Sum of a Function and a Constant. 12. If u = y + c, where y represents any function of x, and c denotes a constant quantity, then

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Let a', y', u', be simultaneous values of x, y, U; then

U = y + C,

u = y' + C, ri y y

su dy 8x Сх


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whence, proceeding to the limit,

du dy
dx dx'

or the differential coefficient of the sum of a function and a constant is the same as that of the function alone.

Differentiation of the Product of a Function and a Constant.

13. If u = cy, where c represents a constant quantity and y a function of x; then


dy dx



Let x', y', u', be simultaneous values of x, y, u; then

U = cy,

ui' = cy', u

δu dy x




y' - y

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or the differential coefficient of the product of a function and a constant is equal to the product of the constant and the differential coefficient of the function.

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