Chapter 2: Differentiation
Section 2.6: Derivatives of the Exponential and Logarithmic Functions
In Section 1.1 the transcendental number e was defined as the limit of 1+h1/h, as h→∞. Functions of the form fx=ax are called exponential functions, and when a=e, the function ex 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 1+x h1/h as h→∞. Implementing the exponential e in Maple was also detailed in Section 1.1.
Just as gx=logax is the functional inverse of fx=ax, so too is lnx, the natural log of x, the functional inverse of ex. Table 2.6.1 lists the derivatives of the exponential and logarithmic functions.
Table 2.6.1 Derivatives of the exponential and logarithmic functions
Th astute reader will note that the derivative of ex is itself. This is the only function for which f′=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.
Table 2.6.2 The technique of logarithmic differentiation
Example 2.6.5 illustrates logarithmic differentiation for a product of three factors.
Derivative of the Natural Logarithm
Table 2.6.4 contains a derivation of ddxlnx=1x, the rule for differentiating the natural logarithm.
Table 2.6.4 Derivation of the differentiation rule for 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 1+x h1/h in Section 1.1. The equivalent result with x→1/x is invoked here.
Derivative of the Exponential Function
Since ex is the functional inverse of lnx, the simplest way to obtain its differentiation rule is to differentiate the identity lnex=x by the Chain rule. This gives
from which it follows that ddxex=ex.
Use logarithmic differentiation to obtain the derivative of Fx=ux⋅vx⋅wx.
Use logarithmic differentiation to obtain the derivative of yx=ax.
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