. Applied calculus; principles and applications . , , )dy -\r {uy , , , )dz+ * DERIVATION OF [I] 39 The differential of the product of any number of variables is the sum of the products of the differential of each by all the vest .^^, ,(N\ DdN-NdD[VI] dl^^j = ^ The differential of a fraction is the denominator by thedifferential of the numerator minus the numerator by thedifferential of the denominator, divided by the square of thedenominator. nx dx. [VII] d(x^) The differential of a variable with a constant exponent is theproduct of the exponent and the variable with the exponent lessone by t
. Applied calculus; principles and applications . , , )dy -\r {uy , , , )dz+ * DERIVATION OF [I] 39 The differential of the product of any number of variables is the sum of the products of the differential of each by all the vest .^^, ,(N\ DdN-NdD[VI] dl^^j = ^ The differential of a fraction is the denominator by thedifferential of the numerator minus the numerator by thedifferential of the denominator, divided by the square of thedenominator. nx dx. [VII] d(x^) The differential of a variable with a constant exponent is theproduct of the exponent and the variable with the exponent lessone by the differential of the variable. 24. Derivation of [I]. — If ?/ is continuously equal to x,it is evident that y and x must change at equal rates;that is, dy _ dxdx dxdx dy = dx. Since dx 1, the rate of x is the unit rate, so in general the rate of a variablewith respect to itself is unity, or thederivative of / (x), when / (x) is x, isone. Geometrically the locus of y = x is the straight line through origin making angle 4> = j with a; jy-dy tan (^ !^x 1, 1 • A-?/ . ^ ^ ^y dy ^ and smce -r^ is constant, -r^ = ~ = \.Ilx I\x ax dy = dx. For examples of [I], if y^ = 2px, d(y^) = d{2px); or if X2 + ^2 = (j2^ ^ (^2 + ^2) == ^ (^2) = 0. 40 DIFFERENTIAL CALCULUS 25. Derivation of [II]. — By definition the value of aconstant is fixed, therefore the rate of a constant is zero;that is, ^ = 0, .-. da = li y = a, a change in x makes no change in y, hence Ay . Ay . Ay = 0, /. -r^ = 0, and since —■ is constant, Ax dx * ^ Geometrically the slope of 2/ = ot (a Hne parallel to x-axis)is at every point zero. 26. Derivation of [III]. — It is manifest that the rate ofthe sum of v •\- y -\- • • • —2! + cis equal to the sum ofthe rates of its parts, v, y, . . —z and c; that is, d{v -j-y + — z + c) _dv _L_dy ^ ^ _^i^dx dx dx dx dx Multiplying by dx, since dc = 0, the result is [III]. Therule shows that differentials are summed like any otheralgebraic quantities
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