Deriving Exponential Functions

The derivative of 10^x is a constant times 10^x, approximately

(d(10^x))/dx ~ (2.303)10^x.

To see this let Δx be a nonzero infinitesimal. Then

The number st[(10^Δx — 1)/Δx] is a constant which does not depend on x and can be shown to be approximately 2.303.

If we start with a given positive real number b instead of 10, we obtain the exponential function with base b, y = b^x. The derivative of b^x is equal to the constant .st[(b^Δx - 1)/Δx] times b^x. This constant depends on b. The derivative is computed as follows: