In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = ln(e^(θ)/(1+e^θ))

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Rewrite the given function to simplify the expression inside the logarithm: \(y = \ln\left( \frac{e^{\theta}}{1 + e^{\theta}} \right)\).

Use the logarithm property \(\ln\left( \frac{a}{b} \right) = \ln(a) - \ln(b)\) to separate the function into \(y = \ln(e^{\theta}) - \ln(1 + e^{\theta})\).

Simplify \(\ln(e^{\theta})\) using the property \(\ln(e^{x}) = x\), so the function becomes \(y = \theta - \ln(1 + e^{\theta})\).

Differentiate each term with respect to \(\theta\): the derivative of \(\theta\) is 1, and for \(-\ln(1 + e^{\theta})\), apply the chain rule.

For the second term, use the chain rule: \(\frac{d}{d\theta} \left[ -\ln(1 + e^{\theta}) \right] = - \frac{1}{1 + e^{\theta}} \cdot \frac{d}{d\theta} (1 + e^{\theta}) = - \frac{e^{\theta}}{1 + e^{\theta}}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative of the Natural Logarithm Function

The derivative of ln(u), where u is a differentiable function of a variable, is (1/u) times the derivative of u. This rule allows us to differentiate logarithmic expressions by first identifying the inner function and then applying the chain rule.

Chain Rule

The chain rule is used to differentiate composite functions. It states that the derivative of a function composed of another function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Simplification of Exponential Expressions

Simplifying expressions involving exponentials, such as e^θ/(1 + e^θ), helps in easier differentiation. Recognizing that e^θ/(1 + e^θ) can be rewritten or simplified aids in applying derivative rules more efficiently.

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