. An elementary treatise on the differential and integral calculus. + rfy — (z + ffe). (2) Subtracting (1) from (2), we have du = dv + dy — dz, (3) which is the differential required. Therefore, the differential of the algebraic sum ofany number of functions is found by taking the alge-braic sum of their differentials. 15. To Differentiate y = ax ± b. (1) Give to x the infinitesimal increment dx, and let dy bethe corresponding infinitesimal increment of y due to theincrement which x takes. Then (1) becomes y + dy = a (x + dx) ± 0- (2) Subtracting (1) from (2), we get dy = adx, (3) which is the
. An elementary treatise on the differential and integral calculus. + rfy — (z + ffe). (2) Subtracting (1) from (2), we have du = dv + dy — dz, (3) which is the differential required. Therefore, the differential of the algebraic sum ofany number of functions is found by taking the alge-braic sum of their differentials. 15. To Differentiate y = ax ± b. (1) Give to x the infinitesimal increment dx, and let dy bethe corresponding infinitesimal increment of y due to theincrement which x takes. Then (1) becomes y + dy = a (x + dx) ± 0- (2) Subtracting (1) from (2), we get dy = adx, (3) which is the required differential. Hence, the differential of the product of a constantby a variable is equal to the constant multiplied bythe differential of the variable: also, if a constant beconnected with a variable by the sign + or —, it dis-appears in differentiation. This may also be proved geometrically as follows: Let AB (Fig. 2) be the line whose equation is y = ax + b,and let (x, y) be any point P on this line. Give OM (= x) 18 GEOMETRIC ILL USTRA Fig. 2 the infinitesimal increment MM (=dx), then the cor-responding increment of MP (=y)will be CP (= dy). Now in the tri-angle OPP we have CF = CPtanCPP;* or letting a = tan CPP, and substi-tuting for CP and CP their values dyand dx9 we have, dy = adx. It is evident that the constant b will disappear in differentiation,from the very nature of constants, which do not admit of increase, andtherefore can take no increment. 16. Differentiation of the Product of two Func-tions. Let u = yz, (1) where y and z are both functions of x. Give x the infini-tesimal increment dx, and let du, dy, dz be the correspond-ing increments of u, y, and z, due to the increment whichstakes. Then (1) becomes u -f du = (y + dy) (z + dz) = yz + zdy -f- ydz + dz dy. (2) Subtracting (1) from (2), and omitting dzdy, since it isan infinitesimal of the second order, and added to others ofthe first order (Art. 11), we have du = zdy + ydz, (3) which i
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