In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
29. y = ln(1/(x√(x+1)))

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Rewrite the function to simplify differentiation. Given \( y = \ln\left( \frac{1}{x\sqrt{x+1}} \right) \), express it as \( y = \ln(1) - \ln\left(x\sqrt{x+1}\right) \). Since \( \ln(1) = 0 \), this simplifies to \( y = -\ln\left(x\sqrt{x+1}\right) \).

Use the property of logarithms to separate the terms inside the logarithm: \( \ln\left(x\sqrt{x+1}\right) = \ln(x) + \ln\left( (x+1)^{1/2} \right) = \ln(x) + \frac{1}{2} \ln(x+1) \). So, \( y = -\left( \ln(x) + \frac{1}{2} \ln(x+1) \right) \).

Differentiate \( y \) with respect to \( x \) by applying the derivative of logarithmic functions: \( \frac{d}{dx} \ln(x) = \frac{1}{x} \) and \( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} \).

Apply the chain rule and linearity of differentiation: \( \frac{dy}{dx} = -\left( \frac{1}{x} + \frac{1}{2} \cdot \frac{1}{x+1} \right) \).

Combine the terms to write the derivative as a single expression, if desired, by finding a common denominator or leaving it as a sum of fractions.

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주요 개념

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Logarithmic Differentiation

Logarithmic differentiation involves applying the properties of logarithms to simplify the differentiation of complex functions, especially those involving products, quotients, or powers. It allows rewriting the function as a sum or difference of simpler logarithmic terms, making the derivative easier to find.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with 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.

Derivative of Logarithmic Functions

The derivative of the natural logarithm function ln(u) with respect to x is (1/u) times the derivative of u. This rule is essential when differentiating functions expressed as logarithms, especially after simplifying the original function using logarithmic properties.

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.29. y = ln(1/(x√(x+1)))