. Differential and integral calculus, an introductory course for colleges and engineering schools. liters of water at thestart, and suppose the rate of flow through each pipe to be 2 per is the amount of water in R after 1 hour? after 2 hours? Ans. liters. liters. 73. Differentials. Definition. The product of the first derivative of a function by theincrement of its argument is defined to be the differential of the func-tion. The differential symbol is d and the differential of f(x) is writtendf(x). Our definition may now be expressed as follows: (1) df(x)=f(x)Ax
. Differential and integral calculus, an introductory course for colleges and engineering schools. liters of water at thestart, and suppose the rate of flow through each pipe to be 2 per is the amount of water in R after 1 hour? after 2 hours? Ans. liters. liters. 73. Differentials. Definition. The product of the first derivative of a function by theincrement of its argument is defined to be the differential of the func-tion. The differential symbol is d and the differential of f(x) is writtendf(x). Our definition may now be expressed as follows: (1) df(x)=f(x)Ax or Df(x)Ax. 98 DIFFERENTIAL CALCULUS §73 If fix) is x itself, we have dx = xAx or DxAx,or, since x = Dx = 1, dx = Ax, that is, the differential of the argument is identical with the incrementof the argument. Writing dx for Ax in (1), we have(2) df(x)=f(x)dx or Df(x)dx* If y be the functional symbol instead of fix), we have(2r) dy = ydx or Dy dx* Hence in the definition the word increment may be replaced bythe word differential. There is a very simple geometric representation of df(x) or &x=dx r TA / A ]df(x)= dy i> Ax=dx B //fix) =y 0 Ap &x~dx X In the right triangle PBT, BT = PB tan PB = Ax = dx and tan 6 = f(x) or y. BT = f{x)dx or ydx,that is, BT = df(x) or dy. Hence, differential f{x) is the increment of the ordinate of the tangent to the graph of f(x) at the point (x, fix)), dx can be chosen at will. It is manifest, then, that the differential of the function is quite different from the increment of the function. * Because of these relations the derivative was formerly termed the differ-ential coefficient, a cumbersome designation that is rapidly falling into disuse. §73 DIFFERENTIALS 99 If formulae (2) and (2) be divided by dx, there results (3) ^sr-rM-AK*) and &mtfmDy. In the first members of these equations we have a new notation forthe derivative. In any formula involving derivatives of functions of x such as Du, Dv, we may replace the latter by -=-, -
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