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

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

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 \(u = \tan x\). Then, find the derivative of \(u\) with respect to \(x\): \(\frac{du}{dx} = \sec^2 x\).

Apply the chain rule by substituting \(u\) and \(\frac{du}{dx}\) into the derivative formula: \(\frac{dy}{dx} = \frac{1}{\sqrt{1 + (\tan x)^2}} \cdot \sec^2 x\).

Simplify the expression if possible, using trigonometric identities such as \(1 + \tan^2 x = \sec^2 x\), to write the derivative in a simpler form.

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Key Concepts

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

Inverse Hyperbolic Functions

Inverse hyperbolic functions, like sinh⁻¹(x), are the inverses of hyperbolic functions. The function sinh⁻¹(x) returns the value whose hyperbolic sine is x. Understanding their definitions and properties is essential for differentiating expressions involving these functions.

Chain Rule

The chain rule is a differentiation technique used when a function is composed of another function, such as y = sinh⁻¹(tan x). 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 Inverse Hyperbolic Sine

The derivative of sinh⁻¹(u) with respect to u is 1/√(1 + u²). This formula is crucial when differentiating y = sinh⁻¹(tan x), as it provides the derivative of the outer function before applying the chain rule to the inner function tan x.

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