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

Verified step by step guidance

1. Identify the function to differentiate: \( r = \sin(\sqrt{2\theta}) \). This is a composition of functions, involving a sine function and a square root function.
2. Apply the chain rule for differentiation, which states that the derivative of a composite function \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \). Here, \( f(x) = \sin(x) \) and \( g(x) = \sqrt{2\theta} \).
3. Differentiate the outer function \( f(x) = \sin(x) \) with respect to its argument: \( f'(x) = \cos(x) \).
4. Differentiate the inner function \( g(x) = \sqrt{2\theta} \) with respect to \( \theta \). First, express \( \sqrt{2\theta} \) as \( (2\theta)^{1/2} \) and use the power rule: \( \frac{d}{d\theta}(2\theta)^{1/2} = \frac{1}{2}(2\theta)^{-1/2} \cdot 2 \).
5. Combine the results using the chain rule: \( \frac{dr}{d\theta} = \cos(\sqrt{2\theta}) \cdot \frac{1}{2}(2\theta)^{-1/2} \cdot 2 \). Simplify the expression to find the derivative.

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Chain Rule

The chain rule is a formula for computing the derivative of a composite function. If a function is composed of two or more functions, the chain rule states that the derivative of the outer function is multiplied by the derivative of the inner function. This is particularly useful when dealing with functions that involve nested expressions, such as trigonometric functions of other functions.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate angles to ratios of sides in right triangles. In calculus, these functions are essential for modeling periodic phenomena and are often involved in differentiation and integration. Understanding their properties and derivatives is crucial for solving problems involving angles and oscillatory behavior.