Chapter 4, Problem 10RE

To determine

To Find: To find the derivative of the given function.

Answer to Problem 10RE

The derivative is $-4\theta {\sec }^2(3-{\theta }^2)\tan (3-{\theta }^2)$ .

Explanation of Solution

Given information: Given function $r={\tan }^2(3-{\theta }^2)$

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

Calculation:

  $\begin{array}{l}r={\tan }^2(3-{\theta }^2)\\ \text{Differentiate}\text{with}\text{respect}\text{to}\theta \\ \frac{dr}{d\theta }=\frac{d{\tan }^2(3-{\theta }^2)}{d\tan (3-{\theta }^2)}\times \frac{d\tan (3-{\theta }^2)}{d(3-{\theta }^2)}\times \frac{d(3-{\theta }^2)}{d\theta }\\ \frac{dr}{d\theta }=2\tan (3-{\theta }^2)\times {\sec }^2(3-{\theta }^2)\times -2\theta \\ \frac{dr}{d\theta }=-4\theta {\sec }^2(3-{\theta }^2)\tan (3-{\theta }^2)\end{array}$