Chapter 4.1, Problem 27E

To determine

To calculate:The differentiation of the given function.

Answer to Problem 27E

The differentiation is $\frac{\theta \times \cos \theta +\sin \theta }{2\sqrt{\theta \sin \theta }}$ .

Explanation of Solution

Given information: Given function is $r=\sqrt{\theta \times \sin \theta }$ .

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

Calculation:

  $r=\sqrt{\theta \sin \theta }$

Differentiate with respect to $\theta$ ,

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