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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 ${\left(1\+xh\right)}^{1sol;h}$ as $h\to \infty$. Implementing the exponential e in Maple was also detailed in Section 1.1.
Just as $g\left(x\right)\={\mathrm{log}}_{a}\left(x\right)$ is the functional inverse of $f\left(x\right)\={a}^{x}$, so too is $\mathrm{ln}\left(x\right)$, the natural log of $x$, the functional inverse of ${e}^{x}$. Table 2.6.1 lists the derivatives of the exponential and logarithmic functions.
Function

Derivative

Example

${e}^{x}$

${e}^{x}$

Example 2.6.1

${a}^{x}$

${a}^{x}\mathrm{ln}\left(a\right)$

Example 2.6.2

$\mathrm{ln}\left(x\right)$

$\frac{1}{x}$

Example 2.6.3

${\mathrm{log}}_{a}\left(x\right)$

$\frac{1}{x\mathrm{ln}\left(a\right)}$

Example 2.6.4

Table 2.6.1 Derivatives of the exponential and logarithmic functions



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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.
Logarithmic Differentiation

$F\left(x\right)\={g}_{1}\left(x\right)\cdot {g}_{2}\left(x\right)\cdot \dots \cdot {g}_{n}\left(x\right)$ $\mathbf{\Rightarrow}$$F\prime \left(x\right)\=F\left(x\right)\cdot \left(\frac{{g}_{1}^{\mathbf{\/}}\left(x\right)}{{g}_{1}\left(x\right)}\+\cdots \+\frac{{g}_{n}^{\mathbf{\/}}\left(x\right)}{{g}_{n}\left(x\right)}\right)$

Table 2.6.2 The technique of logarithmic differentiation



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