In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
25. y = sinh⁻¹(√x)

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Identify the function: \( y = \sinh^{-1}(\sqrt{x}) \). Here, \( y \) is the inverse hyperbolic sine of \( \sqrt{x} \).

Recall the derivative formula for the inverse hyperbolic sine function: \( \frac{d}{dx} \sinh^{-1}(u) = \frac{1}{\sqrt{1+u^2}} \cdot \frac{du}{dx} \), where \( u \) is a function of \( x \).

Set \( u = \sqrt{x} = x^{1/2} \). Then find \( \frac{du}{dx} \) using the power rule: \( \frac{du}{dx} = \frac{1}{2} x^{-1/2} \).

Apply the chain rule: \( \frac{dy}{dx} = \frac{1}{\sqrt{1 + (\sqrt{x})^2}} \cdot \frac{1}{2} x^{-1/2} \).

Simplify the expression inside the square root: \( (\sqrt{x})^2 = x \), so the derivative becomes \( \frac{dy}{dx} = \frac{1}{\sqrt{1 + x}} \cdot \frac{1}{2} x^{-1/2} \).

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Key Concepts

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

Inverse Hyperbolic Sine Function (sinh⁻¹)

The inverse hyperbolic sine function, sinh⁻¹(x), is the inverse of the hyperbolic sine function. It can be expressed as sinh⁻¹(x) = ln(x + √(x² + 1)), which helps in differentiating expressions involving sinh⁻¹.

Chain Rule

The chain rule is a differentiation technique used when a function is composed of other functions. 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.

Derivative of Square Root Function

The square root function, √x, can be written as x^(1/2). Its derivative is (1/2)x^(-1/2), which is essential when differentiating expressions where the square root is inside another function.

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