Logarithmic differentiation

Instead of applying (VIII) and (VIIIa) at once in differentiating logarithmic functions, we may sometimes simplify the work by first making use of one of the formulas 7-10 in §1.1. Thus above Illustrative Example 5.15.2 may be solved as follows:

Example 5.16.1   Differentiate $y = \log \sqrt{1 - x^2}$.

Solution. By using 10, in §1.1, we may write this in a form free from radicals as follows: $y = \frac{1}{2} \log (1 - x^2)$. Then

$$\begin{array}{ll} \frac{dy}{dx} &= \frac{1}{2} \frac{\frac... ...}{2} \cdot \frac{-2}{1 - x^2} = \frac{x}{x^2 - 1}. \end{array}$$

Example 5.16.2   Differentiate $y = \log \sqrt{\frac{1 + x^2}{1 - x^2}}$.

Solution. Simplifying by means of 10 and 8, in §1.1,

$$\begin{array}{ll} y &= \frac{1}{2} [ \log (1 + x^2) - \log ... ...1 + x^2} + \frac{x}{1 - x^2} = \frac{2x}{1 - x^4}. \end{array}$$

In differentiating an exponential function, especially a variable with a variable exponent, the best plan is first to take the logarithm of the function and then differentiate. Thus Example 5.15.5 is solved more elegantly as follows:

Example 5.16.3   Differentiate $y = x^{e^x}$.

Solution. Taking the logarithm of both sides, $\log y = e^x\log x$, by 9 in §1.1. Now differentiate both sides with respect to $x$:

$$\begin{array}{ll} \frac{\frac{dy}{dx}}{y} & = e^x \frac{d}{... ...V}\\ & = e^x \cdot \frac{1}{x} + \log x \cdot e^x, \end{array}$$

or,

$$\frac{dy}{dx} = e^x \cdot y \left ( \frac{1}{x} \log x \right ) = e^x x^{e^x} \left ( \frac{1}{x} + \log x \right ).$$

Example 5.16.4   Differentiate $y = (4x^2 - 7)^{2 + \sqrt{x^2 - 5}}$.

Solution. Taking the logarithm of both sides,

$$\log y = (2 + \sqrt{x^2 - 5}) \log (4x^2 - 7).$$

Differentiating both sides with respect to $x$,

$$\frac{1}{y} \frac{dy}{dx} = (2 + \sqrt{x^2 - 5}) \frac{8x}{4x^2 - 7} + \log(4x^2 - 7) \cdot \frac{x}{\sqrt{x^2 - 5}}.$$

$$\frac{dy}{dx} = x(4x^2 - 7)^{2 + \sqrt{x^2 - 5}} \left [ \frac{8(... ...\sqrt{x^2 - 5})}{4x^2 - 7} + \frac{\log (4x^2 - 7)}{\sqrt{x^2 - 5}} \right ].$$

In the case of a function consisting of a number of factors it is sometimes convenient to take the logarithm before differentiating. Thus,

Example 5.16.5   Differentiate $y = \sqrt{\frac{(x - 1)(x - 2)}{(x - 3)(x - 4)}}$.

Solution. Taking the logarithm of both sides,

$$\log y = \frac{1}{2} [\log (x -1) + \log (x - 2) - \log(x - 3) - \log(x - 4)].$$

Differentiating both sides with respect to $x$,

$$\begin{array}{ll} \frac{1}{y} \frac{dy}{dx} &= \frac{1}{2} \... ...rac{2x^2 - 10x + 11}{(x - 1)(x - 2)(x - 3)(x - 4)}, \end{array}$$

or,

$$\frac{dy}{dx} = -\frac{2x^2 - 10x - 11}{(x - 1)^{\frac{1}{2}} (x - 2)^{\frac{1}{2}} ( x - 3)^{\frac{3}{2}} (x - 4)^{\frac{3}{2}}}.$$