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Chapter 12: Problem 25

Find the derivative of the given function.\(h(x)=\cosh ^{-1}(\csc x)\)

Short Answer

Expert verified
The derivative is \(-\text{csc} x\).

Step by step solution

01

- Understand the given function

The given function is \(h(x) = \text{cosh}^{-1}(\text{csc } x)\). The goal is to find the derivative of this function with respect to \(x\).
02

- Recall the derivative of the inverse hyperbolic cosine function

The derivative of \(\text{cosh}^{-1}(u)\) with respect to \(u\) is \(\frac{1}{\sqrt{u^2 - 1}}\).
03

- Differentiate the composite function

To find the derivative of \(h(x) = \text{cosh}^{-1}(\text{csc } x)\), use the chain rule. First, let \(u = \text{csc } x\). Then \(h(x) = \text{cosh}^{-1}(u)\). Applying the chain rule, \(\frac{dh}{dx} = \frac{dh}{du} \frac{du}{dx}\).
04

- Compute \(\frac{du}{dx}\) for \(u = \text{csc } x\)

The derivative of \(\text{csc } x\) is \(-\text{csc } x \cot x\). Therefore, \(\frac{du}{dx} = -\text{csc } x \cot x\).
05

- Combine results using the chain rule

Using the chain rule, \(\frac{dh}{dx} = \frac{1}{\sqrt{u^2 - 1}} \cdot \frac{du}{dx}\). Substituting \(u = \text{csc } x\) and \(\frac{du}{dx}\), we get \(\frac{dh}{dx} = \frac{1}{\sqrt{\text{csc}^2 x - 1}} \cdot (-\text{csc } x \cot x)\).
06

- Simplify the expression

Recall that \(\text{csc}^2 x - 1 = \cot^2 x\). Therefore, \(\frac{1}{\sqrt{\text{csc}^2 x - 1}} = \frac{1}{\sqrt{\cot^2 x}} = \frac{1}{|\cot x|}\). Since \(|\cot x| = \cot x\) for \(0 < x < \pi\), the expression simplifies to \(\frac{dh}{dx} = -\frac{\text{csc } x \cot x}{\cot x} = -\text{csc } x\).

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Key Concepts

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