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

\begin{array}{c} x^y = z \\ \hline \text{Then will } y \dot{z} = \dot{z} \\ \hline l \times dy + \frac{y dx}{x} = \frac{dz}{z} \\ \hline l \times dy + \frac{xy dx}{x} = dz \end{array}

That is, x^y l \times dy + x^y \dot{x} = dz.