In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
27. y = (((t+1)(t-1))/((t-2)(t+3)))^5, t>2

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Start by taking the natural logarithm of both sides of the equation to simplify the differentiation process: \(\ln y = \ln \left( \left( \frac{(t+1)(t-1)}{(t-2)(t+3)} \right)^5 \right)\).

Use the logarithm power rule to bring down the exponent: \(\ln y = 5 \ln \left( \frac{(t+1)(t-1)}{(t-2)(t+3)} \right)\).

Apply the logarithm quotient rule to separate the numerator and denominator inside the logarithm: \(\ln y = 5 \left( \ln (t+1) + \ln (t-1) - \ln (t-2) - \ln (t+3) \right)\).

Differentiate both sides with respect to \(t\), remembering that \(\frac{d}{dt} (\ln y) = \frac{1}{y} \frac{dy}{dt}\) by implicit differentiation: \(\frac{1}{y} \frac{dy}{dt} = 5 \left( \frac{1}{t+1} + \frac{1}{t-1} - \frac{1}{t-2} - \frac{1}{t+3} \right)\).

Finally, solve for \(\frac{dy}{dt}\) by multiplying both sides by \(y\): \(\frac{dy}{dt} = y \cdot 5 \left( \frac{1}{t+1} + \frac{1}{t-1} - \frac{1}{t-2} - \frac{1}{t+3} \right)\), and substitute back the original expression for \(y\) if desired.

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주요 개념

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Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products, quotients, or powers of other functions. By taking the natural logarithm of both sides, the differentiation process simplifies using properties of logarithms, especially useful for complicated expressions raised to a power.

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 complex expressions to be broken down into simpler sums and differences. These properties are essential in logarithmic differentiation to simplify the function before differentiating.

Chain Rule

The chain rule is a fundamental differentiation rule used when differentiating composite functions. In logarithmic differentiation, after taking the logarithm, the derivative of the outer function (logarithm) is combined with the derivative of the inner function, enabling the calculation of the original function's derivative.

In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.27. y = (((t+1)(t-1))/((t-2)(t+3)))^5, t>2