Find the derivatives of the functions in Exercises 1–42.
___
𝓻 = sin √ 2θ

Verified step by step guidance

1. Identify the function: The given function is \( r = \sin(\sqrt{2\theta}) \). This is a composition of functions, where the outer function is \( \sin(u) \) and the inner function is \( u = \sqrt{2\theta} \).
2. Apply the chain rule: The chain rule states that the derivative of a composite function \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \). Here, \( f(u) = \sin(u) \) and \( g(\theta) = \sqrt{2\theta} \).
3. Differentiate the outer function: The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \). So, \( f'(u) = \cos(u) \).
4. Differentiate the inner function: The derivative of \( \sqrt{2\theta} \) with respect to \( \theta \) is \( \frac{d}{d\theta}(2\theta)^{1/2} \). Use the power rule: \( \frac{1}{2}(2\theta)^{-1/2} \cdot 2 = (2\theta)^{-1/2} \).
5. Combine the derivatives: Multiply the derivative of the outer function by the derivative of the inner function: \( \cos(\sqrt{2\theta}) \cdot (2\theta)^{-1/2} \). This is the derivative of the original function \( r = \sin(\sqrt{2\theta}) \) with respect to \( \theta \).