In Section 1.1 the transcendental number $e$ was defined as the limit of ${\left(1\+h\right)}^{1\/h}$, as $h\to \infty$. Functions of the form $f\left(x\right)\=a^x$ are called exponential functions, and when $a\=e$, the function $e^x$ is called the exponential function. This function can be realized as an exponential function with $a\=e$, or, as seen in Section 1.1, it can be realized as the limit of ${\left(1\+x h\right)}^{1\/h}$ as $h\to \infty$. Implementing the exponential e in Maple was also detailed in Section 1.1.

Just as $g\left(x\right)\={\log }_a\left(x\right)$ is the functional inverse of $f\left(x\right)\=a^x$, so too is $\ln \left(x\right)$, the natural log of $x$, the functional inverse of $e^x$. Table 2.6.1 lists the derivatives of the exponential and logarithmic functions.

Th astute reader will note that the derivative of $e^x$ is itself. This is the only function for which $f\prime \=f$, that is, for which the derivative equals the function itself. Moreover, the astute reader will note that the derivative of the natural log function is a power of $x$. In other words, differentiating this particular transcendental function results in a rational power of $x$.

The technique of logarithmic differentiation for the derivative of a product of multiple factors is detailed in Table 2.6.2.

Example 2.6.5 illustrates logarithmic differentiation for a product of three factors.

Table 2.6.4 contains a derivation of $\frac{d}{\mathrm{dx}}\ln \left(x\right)\=\frac{1}{x}$, the rule for differentiating the natural logarithm.

The interchange of the limit with the logarithm in the fourth line is a result of the continuity of the logarithm. The resulting limit has been articulated for ${\left(1\+x h\right)}^{1\/h}$ in Section 1.1. The equivalent result with $x\to 1\/x$ is invoked here.

Since $e^x$ is the functional inverse of $\ln \left(x\right)$, the simplest way to obtain its differentiation rule is to differentiate the identity $\ln \left(e^x\right)\=x$ by the Chain rule. This gives

$\frac{1}{e^x} \frac{d}{\mathrm{dx}}\left(e^x\right)\=1$ 

from which it follows that $\frac{d}{\mathrm{dx}}\left(e^x\right)\=e^x$.