In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
29. y = (sin θ)^√θ

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Start by taking the natural logarithm of both sides of the equation to simplify the differentiation process: \(\ln y = \ln \left( (\sin \theta)^{\sqrt{\theta}} \right)\).

Use the logarithm power rule to bring the exponent down: \(\ln y = \sqrt{\theta} \cdot \ln (\sin \theta)\).

Differentiate both sides with respect to \(\theta\). Remember to use implicit differentiation on the left side and the product rule on the right side: \(\frac{1}{y} \frac{dy}{d\theta} = \frac{d}{d\theta} \left( \sqrt{\theta} \cdot \ln (\sin \theta) \right)\).

Apply the product rule to the right side: \(\frac{d}{d\theta} \left( \sqrt{\theta} \right) \cdot \ln (\sin \theta) + \sqrt{\theta} \cdot \frac{d}{d\theta} \left( \ln (\sin \theta) \right)\).

Calculate each derivative separately: \(\frac{d}{d\theta} \left( \sqrt{\theta} \right) = \frac{1}{2 \sqrt{\theta}}\) and \(\frac{d}{d\theta} \left( \ln (\sin \theta) \right) = \frac{\cos \theta}{\sin \theta} = \cot \theta\). Then substitute back into the expression and multiply both sides by \(y\) to solve for \(\frac{dy}{d\theta}\).

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Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions where the variable appears both in the base and the exponent. By taking the natural logarithm of both sides, the expression simplifies, allowing the use of implicit differentiation and the properties of logarithms to find the derivative more easily.

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Logarithmic Differentiation

Chain Rule

The chain rule is a fundamental differentiation rule used when differentiating composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This is essential when differentiating expressions like sin(θ) raised to a function of θ.

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Intro to the Chain Rule

Properties of Logarithms

Properties of logarithms, such as log(a^b) = b log(a), allow us to simplify complex expressions involving exponents. Applying these properties helps transform the original function into a form that is easier to differentiate, especially when the exponent itself is a function of the variable.

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