y will continuously vary from one value to another as a decreases from any assigned positive value down to zero, but, the moment becomes negative, y becomes impossible. The two branches corresponding to the double sign, each of which terminates abruptly at the origin, join together at this point and thus form a continuous curve. CHAPTER II. PRINCIPLES OF DIFFERENTIATION. SECTION I. GENERAL FUNCTIONS. 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 x' keeps continuously approaching to the value of x, the fraction 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 x - x, y' y, are usually denoted by the symbols Sx, Sy, and the ultimate value of the fraction In this expression dx and dy are any quantities whatever, either finite or infinitesimal, which are in the ratio of the dy ultimate values of Sx and dy. The fraction is called the dx differential coefficient of y with regard to x, the quantities dx and dy being called the differentials of x and y. The object of the Differential Calculus is to investigate the pro perties 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 y3: then y' = x"3: whence = approaches indefinitely near to x, the left-hand When that is, the differential coefficient of x3 with respect to x is 3x2, and its differential is 3x2 dx. Differentiation of a Constant. 11. If y = c, where c denotes any constant quantity, that is, any quantity which does not experience variation in consequence of a variation in the value of x, then or the differential coefficient of a constant quantity is always zero. 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 du dy Let x', y', u', be simultaneous values of x, y, u; then 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 Let x', y', u', be simultaneous values of x, y, u; then 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. In fact, taking any simultaneous values x', u', y', y'29 Y's, ... Y'„› Hence the differential coefficient of the sum of any number of functions is equal to the sum of the differential coefficients of the functions taken separately. Differentiation of the Product of two Functions. 15. If u = y1y2, where y, and y, are any functions of x, then 2 We have x', u', y', y', denoting simultaneous values of |
