Chapter 4, Problem 18RE

To determine

To Find: The derivative of the given function.

Answer to Problem 18RE

The derivative is $\frac{2}{\theta \log 2}$ .

Explanation of Solution

Given information: Given function $r={\log }_2({\theta }^2)$

Formula used: $\frac{d(\log x)}{dx}=\frac{1}{x}\text{,}\frac{d(x^n)}{dx}=n{x}^{n-1}\text{and}{\log }_{b}a=\frac{\log a}{\log b}$ with chain rule.

Calculation:

  $\begin{array}{l}r={\log }_2({\theta }^2)\\ r=\frac{\log ({\theta }^2)}{\log 2}\\ \text{Differentiate}\text{with}\text{respect}\text{to}\theta \\ \frac{dr}{d\theta }=\frac{1}{\log 2}\times \frac{d\log ({\theta }^2)}{d({\theta }^2)}\times \frac{d({\theta }^2)}{d\theta }\\ \frac{dr}{d\theta }=\frac{1}{\log 2}\times \frac{1}{{\theta }^2}\times 2\theta \\ \frac{dr}{d\theta }=\frac{2}{\theta \log 2}\end{array}$