CALCULUS Exponentialis

CALCULUS Exponentialis, is a method of differencing exponential quantities, or of finding and summing up the differentials or moments of exponential quantities; or at least bringing them to geometrical constructions.

By exponential quantity, is here understood a power, whose exponent is variable; v. g. $x^x$, $a^x$, $x^y$, where the exponent $x$ does not denote the same in all the points of a curve, but in some stands for 2, in others for 3, in others for 5, &c.

To difference an exponential quantity; there is nothing required but to reduce the exponential quantities to logarithmic ones; which done, the differencing is managed as in logarithmic quantities. Thus, suppose the differential of the exponential quantity $x^y$ required, let

Then will $y/x\; =\; l/z$

That is, $x^y\; l\; x\; dy\; +\; x^y\; \backslash frac\{y\; dx\}\{x\}\; =\; dz$.