Elements of analysis as applied to the mechanics of engineering and machinery . M N X A. M N This is the most general rule for the determination of the differ-ential of a function, from which, by application to different functions,other rules more or less general may be deduced. If we have, for instance, y = ^^, then dy = (x-{-dxy — ^^,or, since we have to put (x -|- dxy = x^-\- 2xdx -J- ^^^there will result dy = 2xdx + dx = (2x + 8^) dx; c 6 ELEMENTS OF ANALYSIS. [Art. 6. or more simply, since dx, as infinitesimal, vanishes when comparedwith 2 a;: dy = d{xy = 2xdoc, y=:x^ corresponds to the a
Elements of analysis as applied to the mechanics of engineering and machinery . M N X A. M N This is the most general rule for the determination of the differ-ential of a function, from which, by application to different functions,other rules more or less general may be deduced. If we have, for instance, y = ^^, then dy = (x-{-dxy — ^^,or, since we have to put (x -|- dxy = x^-\- 2xdx -J- ^^^there will result dy = 2xdx + dx = (2x + 8^) dx; c 6 ELEMENTS OF ANALYSIS. [Art. 6. or more simply, since dx, as infinitesimal, vanishes when comparedwith 2 a;: dy = d{xy = 2xdoc, y=:x^ corresponds to the area of a square AB GD^ Fig. 11, whose side AB ^^ AD is = ^; and we may infer from ^^ the figure, that, by the augmentation of the sides N PQ lojBM=DJSf=dx^ the square is enlarged by _i j_Q two rectangles BO and DF ^ 2xdT. and a square OF = (dxY; that, therefore, with an in-finitesimal increment (dx) of x^ the square y = x^increases by the element 2xdx.^~M Art. 6. The straight line TF Q, Figs. 9 and 10, which passes through two infinitely approx-imate points P, Q^ of a curve, is cal
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