Question

In Exercises \(25-28\) find \(d r / d \theta\). $$r=\sqrt{\theta \sin \theta}$$

Short answer

The derivative of the function \(r = \sqrt{\theta \sin \theta}\) with respect to \(\theta\) is \(d r / d \theta = \frac{1}{2} (\theta \sin \theta)^{-\frac{1}{2}} (\theta \cos \theta + \sin \theta)\)

Step-by-step solution

Simplify the Function Expression

Consider the function \(r = \sqrt{\theta \sin \theta}\). To simplify the process of finding its derivative, it can be rewritten as: \(r = (\theta \sin \theta)^\frac{1}{2}\)

Apply the Chain Rule

The chain rule states that the derivative of a composition of functions is the derivative of the outer function times the derivative of the inner function. Applying the chain rule and power rule (which states that the derivative of \(x^n\) is \(n \cdot x^{n - 1}\)), the derivative \(d r / d \theta\) is: \(\frac{1}{2} (\theta \sin \theta)^{\frac{1}{2} - 1} \cdot d/d\theta (\theta \sin \theta)\)

Differentiate the Inner Function

The derivative of the inner function \(\theta \sin \theta\) can be calculated by using the product rule. The product rule states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Thus, \(d/d\theta (\theta \sin \theta) = \theta \cdot \cos \theta + \sin \theta\)

Substitute and Simplify

Substitute the calculated derivative of the inner function into the equation from Step 2. Thus, \(d r / d \theta = \frac{1}{2} (\theta \sin \theta)^{-\frac{1}{2}} (\theta \cos \theta + \sin \theta)\)