In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
25. y=arcsec(2s+1)

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Identify the function given: \(y = \arcsec(2s + 1)\). We need to find \(\frac{dy}{ds}\), the derivative of \(y\) with respect to \(s\).

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

Set \(u = 2s + 1\). Compute the derivative of \(u\) with respect to \(s\): \(\frac{du}{ds} = 2\).

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

Simplify the expression inside the square root and write the final derivative expression in terms of \(s\), leaving it in simplified radical form without calculating the numeric value.

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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 arcsec(x) have specific derivative formulas. Understanding these formulas is essential to differentiate functions involving inverse trig expressions correctly.

Chain Rule

The chain rule is used to differentiate composite functions. When the argument of arcsec is a function of x, such as 2s+1, the derivative of the outer function is multiplied by the derivative of the inner function.

Domain and Differentiability of arcsec(x)

The function arcsec(x) is defined for |x| ≥ 1, and its derivative involves expressions with sqrt(x^2 - 1). Recognizing the domain restrictions and the form of the derivative helps avoid errors in differentiation.

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.25. y=arcsec(2s+1)