In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
5. y = ln(sin²θ)

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Recognize that the function is a composition of functions: \(y = \ln(\sin^{2}\theta)\), which can be rewritten as \(y = \ln((\sin\theta)^{2})\).

Use the logarithmic property to simplify the expression: \(\ln(a^{b}) = b \ln(a)\), so rewrite \(y\) as \(y = 2 \ln(\sin\theta)\).

Differentiate \(y\) with respect to \(\theta\) using the chain rule. The derivative of \(\ln(u)\) with respect to \(\theta\) is \(\frac{1}{u} \cdot \frac{du}{d\theta}\), where \(u = \sin\theta\).

Calculate \(\frac{du}{d\theta}\) where \(u = \sin\theta\), so \(\frac{du}{d\theta} = \cos\theta\).

Combine the results to write the derivative: \(\frac{dy}{d\theta} = 2 \cdot \frac{1}{\sin\theta} \cdot \cos\theta\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Chain Rule

The chain rule is a method for finding the derivative of a composite function. It states that the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x). This rule is essential when differentiating functions like ln(sin²θ), where one function is nested inside another.

Derivative of the Natural Logarithm Function

The derivative of ln(u), where u is a differentiable function of x, is (1/u) times the derivative of u. This property allows us to differentiate logarithmic functions by first identifying the inner function and then applying the chain rule.

Derivative of Trigonometric Functions

Understanding the derivatives of basic trigonometric functions is crucial. For example, the derivative of sin(θ) with respect to θ is cos(θ). When differentiating sin²θ, the chain rule applies again, treating sin²θ as (sin θ)².

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