In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
23. y = arccsc(secθ), 0<θ<π/2

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Recognize that the function is given as \(y = \arccsc(\sec \theta)\), where \(0 < \theta < \frac{\pi}{2}\). Our goal is to find \(\frac{dy}{d\theta}\).

Recall the derivative formula for the inverse cosecant function: if \(y = \arccsc(u)\), then \(\frac{dy}{dx} = -\frac{1}{|u| \sqrt{u^2 - 1}} \cdot \frac{du}{dx}\), where \(u\) is a function of \(x\).

Identify the inner function \(u = \sec \theta\). Next, find the derivative of \(u\) with respect to \(\theta\): \(\frac{du}{d\theta} = \sec \theta \tan \theta\).

Substitute \(u = \sec \theta\) and \(\frac{du}{d\theta} = \sec \theta \tan \theta\) into the derivative formula for \(\arccsc(u)\):

\[\frac{dy}{d\theta} = -\frac{1}{|\sec \theta| \sqrt{(\sec \theta)^2 - 1}} \cdot \sec \theta \tan \theta.\]

Simplify the expression inside the square root using the identity \(\sec^2 \theta - 1 = \tan^2 \theta\), and simplify the entire expression accordingly to express \(\frac{dy}{d\theta}\) in terms of \(\theta\).

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Key Concepts

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

Inverse Trigonometric Functions

Inverse trigonometric functions, like arccsc and sec, reverse the effect of their corresponding trigonometric functions. Understanding their domains and ranges is essential, as well as knowing how to express and differentiate them. For example, arccsc(x) is the inverse of csc(x), defined for |x| ≥ 1.

Chain Rule

The chain rule is a differentiation technique used when a function is composed of another function, such as y = arccsc(secθ). 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.

Derivatives of Inverse Trigonometric Functions

Each inverse trig function has a specific derivative formula, for example, d/dx[arccsc(x)] = -1 / (|x|√(x² - 1)). Knowing these formulas allows you to differentiate expressions involving inverse trig functions accurately, especially when combined with the chain rule.

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