Chapter 4, Problem 6RE

To determine

To Find: The derivative of the given function.

Answer to Problem 6RE

The derivative is $\frac{2}{t^2}\cos e{c}^2\left(\frac{2}{t}\right)$ .

Explanation of Solution

Given information: Given function $s=\cot \left(\frac{2}{t}\right)$

Formula used: $\frac{d(\frac{1}{x})}{dx}=\frac{-1}{x^2}\text{and}\frac{d(\cot x)}{dx}=-\cos e{c}^{2}x$ with chain rule.

Calculation:

  $\begin{array}{l}s=\cot \left(\frac{2}{t}\right)\\ \text{Differentiate}\text{with}\text{respect}\text{to}t\\ \frac{ds}{dt}=\frac{d\cot \left(\frac{2}{t}\right)}{d\left(\frac{2}{t}\right)}\times \frac{d\left(\frac{2}{t}\right)}{dt}\\ \frac{ds}{dt}=-\cos e{c}^2\left(\frac{2}{t}\right)\times \left(\frac{-2}{t^2}\right)\\ \frac{ds}{dt}=\frac{2}{t^2}\cos e{c}^2\left(\frac{2}{t}\right)\end{array}$