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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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).

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