Chapter 4.1, Problem 28E

To determine

To calculate:The differentiation of the given function.

Answer to Problem 28E

The differentiation is $2\times \sqrt{\sec \theta }\times \left(\frac{\theta \times \tan \theta }{2}+1\right)$ .

Explanation of Solution

Given information: Given function is $r=2\theta \sqrt{\sec \theta }$ .

Formula used: $\frac{d(x^n)}{dx}=n{x}^{n-1}\text{and}\frac{d(\sec x)}{dx}=\sec x\tan x$ with chain and product rule.

Calculation:

  $r=2\times \theta \times \sqrt{\sec \theta }$

Differentiate with respect to $\theta$ :

  $\begin{array}{l}\frac{dr}{d\theta }=2\times \theta \times \frac{d(\sqrt{\sec \theta })}{d(\sec \theta )}\times \frac{d(\sec \theta )}{d\theta }+2\times \sqrt{\sec \theta }\times \frac{d\theta }{d\theta }\\ \frac{dr}{d\theta }=\frac{2\theta }{2\sqrt{\sec \theta }}\times \sec \theta \times \tan \theta +2\times \sqrt{\sec \theta }\\ \frac{dr}{d\theta }=2\times \sqrt{\sec \theta }\times \left(\frac{\theta \times \tan \theta }{2}+1\right)\end{array}$