Chapter 3.4, Problem 16E

To determine

Find the derivative of the given function.

Answer to Problem 16E

The derivative of the given function is $\frac{dy}{d\theta }=-3n{\csc }^2\left(n\theta \right)$ .

Explanation of Solution

Given:

The given function is $y=3\cot \left(n\theta \right)$ .

Calculation:

  $y=3\cot \left(n\theta \right)$

Apply chain rule.

  $\frac{df\left(a\right)}{dx}=\frac{df}{da}\cdot \frac{da}{dx}$

Let $f=\cot a,a=n\theta$

  $\frac{dy}{d\theta }=3\frac{d}{da}\left(\cot a\right)\cdot \frac{d}{d\theta }\left(n\theta \right)$

Use derivative rule.

  $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$ and $\frac{d}{dx}\left(\cot x\right)=-{\csc }^{2}x$

  $\begin{array}{l}\Rightarrow \frac{dy}{d\theta }=3\left(-{\csc }^{2}a\right)\cdot n\\ \Rightarrow \frac{dy}{d\theta }=-3n{\csc }^2\left(a\right)\end{array}$

Substitute the value of $a=n\theta$ .

  $\Rightarrow \frac{dy}{d\theta }=-3n{\csc }^2\left(n\theta \right)$

Hence the derivativeof the given function is $\frac{dy}{d\theta }=-3n{\csc }^2\left(n\theta \right)$ .