In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
25. y = 2(x² + 1)/√(cos 2x)

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Start by rewriting the function to make logarithmic differentiation easier. Given \( y = \frac{2(x^2 + 1)}{\sqrt{\cos 2x}} \), express it as \( y = 2(x^2 + 1)(\cos 2x)^{-1/2} \).

Take the natural logarithm of both sides: \( \ln y = \ln 2 + \ln (x^2 + 1) + \ln (\cos 2x)^{-1/2} \).

Simplify the logarithmic expression using log properties: \( \ln y = \ln 2 + \ln (x^2 + 1) - \frac{1}{2} \ln (\cos 2x) \).

Differentiate both sides with respect to \( x \). Remember to use the chain rule on \( \ln y \), which gives \( \frac{1}{y} \frac{dy}{dx} \), and apply the derivatives of each term on the right side.

Solve for \( \frac{dy}{dx} \) by multiplying both sides by \( y \), and then substitute back the original expression for \( y \) to express the derivative in terms of \( x \).

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Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products, quotients, or powers of complicated expressions. By taking the natural logarithm of both sides, the differentiation process simplifies using properties of logarithms, such as turning products into sums and powers into multipliers.

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Logarithmic Differentiation

Properties of Logarithms

Understanding the properties of logarithms is essential for logarithmic differentiation. Key properties include log(ab) = log a + log b, log(a/b) = log a - log b, and log(a^n) = n log a. These allow complex expressions to be broken down into simpler parts that are easier to differentiate.

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Change of Base Property

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 crucial when differentiating expressions like cos(2x) inside the logarithm.

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