In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
35. y=arccsc(e^t)

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Identify the function given: \(y = \arccsc(e^{t})\). We want to find \(\frac{dy}{dt}\), the derivative of \(y\) with respect to \(t\).

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

In this problem, \(u = e^{t}\). Compute the derivative of \(u\) with respect to \(t\): \(\frac{du}{dt} = \frac{d}{dt} e^{t} = e^{t}\).

Substitute \(u = e^{t}\) and \(\frac{du}{dt} = e^{t}\) into the derivative formula: \(\frac{dy}{dt} = -\frac{1}{|e^{t}| \sqrt{(e^{t})^{2} - 1}} \cdot e^{t}\).

Simplify the expression where possible, noting that \(|e^{t}| = e^{t}\) since \(e^{t} > 0\) for all real \(t\). This will give the derivative in terms of \(t\).

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Key Concepts

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

Derivative of Inverse Trigonometric Functions

Inverse trigonometric functions like arccsc(x) have specific derivative formulas. For arccsc(x), the derivative is -1 / (|x|√(x² - 1)). Understanding this formula is essential to differentiate expressions involving arccsc.

Chain Rule

The chain rule is used to differentiate composite functions. When y = arccsc(e^t), you differentiate the outer function arccsc(u) with respect to u, then multiply by the derivative of the inner function e^t with respect to t.

Exponential Function Derivative

The derivative of the exponential function e^t with respect to t is e^t itself. Recognizing this allows you to correctly apply the chain rule when differentiating functions like arccsc(e^t).

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