Chapter 4.1, Problem 26E

To determine

To calculate:The differentiation of the given function.

Answer to Problem 26E

The differentiation is $2\times \sec 2\theta \times ({\sec }^{2}2\theta +{\tan }^{2}2\theta )$ .

Explanation of Solution

Given information: Given function is $r=\sec 2\theta \times \tan 2\theta$ .

Formula used: $\frac{d(x^n)}{dx}=n{x}^{n-1},\frac{d(\tan x)}{dx}={\sec }^{2}x\text{and}\frac{d(\sec x)}{dx}=\sec x\tan x$ with chain and product rule.

Calculation:

  $r=\sec 2\theta \times \tan 2\theta$

Differentiate with respect to $\theta$ ,

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