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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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.

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Derivatives of Other Inverse Trigonometric Functions

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.

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Intro to the Chain Rule

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.

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