Chapter 4, Problem 19RE

To determine

To Find: The derivative of the given function.

Answer to Problem 19RE

The derivative is $\frac{1}{(t-7)\log 5}$ .

Explanation of Solution

Given information: Given function $s={\log }_5(t-7)$

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

Calculation:

  $\begin{array}{l}s={\log }_5(t-7)\\ s=\frac{\log (t-7)}{\log 5}\\ \text{Differentiate}\text{with}\text{respect}\text{to}t\\ \frac{ds}{dt}=\frac{1}{\log 5}\times \frac{d\log (t-7)}{d(t-7)}\times \frac{d(t-7)}{dt}\\ \frac{ds}{dt}=\frac{1}{\log 5}\times \frac{1}{(t-7)}\times 1\\ \frac{ds}{dt}=\frac{1}{(t-7)\log 5}\end{array}$