In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
33. y = csch⁻¹(1/2)^θ

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Identify the function given: \(y = \csch^{-1} \left( \frac{1}{2} \right)^\theta\). Notice that \(\csch^{-1} \left( \frac{1}{2} \right)\) is a constant since it does not depend on \(\theta\).

Rewrite the function to clarify the structure: \(y = \left( \csch^{-1} \left( \frac{1}{2} \right) \right)^\theta\). This is an exponential function with a constant base and variable exponent.

Recall the derivative formula for an exponential function with a constant base \(a\) and variable exponent \(x\): \(\frac{d}{dx} a^x = a^x \ln(a)\).

Apply the derivative formula to \(y\): \(\frac{dy}{d\theta} = \left( \csch^{-1} \left( \frac{1}{2} \right) \right)^\theta \cdot \ln \left( \csch^{-1} \left( \frac{1}{2} \right) \right)\).

This expression represents the derivative of \(y\) with respect to \(\theta\). You can leave the answer in this form or evaluate \(\csch^{-1} \left( \frac{1}{2} \right)\) if needed.

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Key Concepts

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

Inverse Hyperbolic Cosecant Function (csch⁻¹)

The inverse hyperbolic cosecant function, denoted csch⁻¹(x), is the inverse of the hyperbolic cosecant function. It returns the value whose hyperbolic cosecant is x. Understanding its definition and domain is essential for differentiating expressions involving csch⁻¹.

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. This is crucial when differentiating functions like csch⁻¹(1/2)^θ.

Derivative of Inverse Hyperbolic Functions

Each inverse hyperbolic function has a specific derivative formula. For csch⁻¹(x), the derivative with respect to x is -1/(|x|√(1+x²)). Knowing this formula allows you to differentiate expressions involving csch⁻¹ accurately.

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In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.

1. lim (x → -2) (x + 2) / (x² - 4)

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Evaluate the integrals in Exercises 33–54.

∫(e^(3x) + 5e^(-x)) dx

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