... In fact, taking any simultaneous values x', u', y',, y',, 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 = Y1Y29 where Y1 and y2 are any functions of x, then dy2 Y1 dx dy, + Y2 Ax We have x', u', y', y', denoting simultaneous values of whence, proceeding to the limit, that is, equating de to zero, and therefore, y, and y, being supposed to be continuous functions of x, putting also dy, and dy, each equal to zero, we have 1 Hence the differential coefficient of the product of two functions is equal to the sum of the products of each function multiplied by the differential coefficient of the other. Taking x', u', y, y, to denote simultaneous values of x, u, y1, y2, we have and therefore, proceeding to the limit, that is, equating x' – x or da to zero, we get, observing that y, becomes y2, Hence, to differentiate the Ratio of two functions, we have the following rule: Multiply the denominator by the differential coefficient of the numerator, and the numerator by the differential coefficient of the denominator: subtract the latter product from the former: this difference divided by the square of the denominator is the differential coefficient of the Ratio. Differentiation of the Product of any number of Functions. 17. If u=y1.y2-y... y, the product of n functions of x, then 2 Y 3 adding these equations together, cancelling terms which are common to both sides of the resulting equation, and observing be a function of x, in which case x will also be then a function of y, Let x', y', be simultaneous values of x, y; then it is evident and that, consequently, proceeding to the limit, Differentiation of a Function of a Function. 19. If u be a function of y, and y a function of x, then that COR. If u be a function of y,, y, of y2, y2 of y,,....and x, it is manifest that we may prove in the same way 20. Let whatever of = f(y1, y1), where f(y,, y,) denotes any function and y, each of the quantities y, and y, being a function of a third quantity x. Let y1, y2, u, become y,', y', u', Now in the limit, when x' differs from x less than by any assignable magnitude, and, first replacing y, by y,, and then y, by y2, dy, dx du dy 2 In this equation it is very important to observe that the numerators of the two fractions although represented by the same symbol du, are essentially different, the numerator of the former corresponding to the ultimate value of the increment and the numerator of the latter to the ultimate value of the increment. |