In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
66. y = θsin(θ)/√(sec(θ))

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Start by rewriting the function to make logarithmic differentiation easier. Given \( y = \frac{\theta \sin(\theta)}{\sqrt{\sec(\theta)}} \), express the denominator with a fractional exponent: \( y = \theta \sin(\theta) \cdot (\sec(\theta))^{-1/2} \).

Take the natural logarithm of both sides to apply logarithmic differentiation: \( \ln y = \ln \theta + \ln \sin(\theta) + \ln (\sec(\theta))^{-1/2} \).

Simplify the logarithm of the power term using the property \( \ln(a^b) = b \ln a \): \( \ln y = \ln \theta + \ln \sin(\theta) - \frac{1}{2} \ln \sec(\theta) \).

Differentiate both sides with respect to \( \theta \), remembering that \( \frac{d}{d\theta} \ln y = \frac{1}{y} \frac{dy}{d\theta} \) by the chain rule. So, \( \frac{1}{y} \frac{dy}{d\theta} = \frac{1}{\theta} + \cot(\theta) - \frac{1}{2} \cdot \frac{d}{d\theta} \ln \sec(\theta) \).

Find the derivative of \( \ln \sec(\theta) \) using the chain rule: \( \frac{d}{d\theta} \ln \sec(\theta) = \tan(\theta) \). Substitute this back into the equation and then solve for \( \frac{dy}{d\theta} \) by multiplying both sides by \( y \).

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Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products, quotients, or powers of variable expressions. By taking the natural logarithm of both sides, the differentiation process simplifies using properties of logarithms, making it easier to handle complex expressions.

Properties of Logarithms

The properties of logarithms, such as log(ab) = log a + log b, log(a/b) = log a - log b, and log(a^n) = n log a, allow us to rewrite complicated products, quotients, and powers into sums and differences. These properties are essential for simplifying the function before differentiating.

Derivative of Trigonometric Functions

Understanding the derivatives of trigonometric functions like sin(θ) and sec(θ) is crucial. For example, d/dθ[sin(θ)] = cos(θ) and d/dθ[sec(θ)] = sec(θ)tan(θ). These derivatives are used after applying logarithmic differentiation to find the final derivative of the given function.

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.66. y = θsin(θ)/√(sec(θ))