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Chapter 6: Problem 8

On what interval is the formula \(d / d x\left(\tanh ^{-1} x\right)=1 /\left(x^{2}-1\right)\) valid?

Short Answer

Expert verified
Answer: The valid interval for the given formula is -1 < x < 1.

Step by step solution

01

Understanding the inverse hyperbolic tangent function

The inverse hyperbolic tangent function, denoted as \(\tanh^{-1}(x)\), can be defined as the inverse function of the hyperbolic tangent function, which is given by \(\tanh(x) = \dfrac{\sinh(x)}{\cosh(x)}\). The domain of the inverse hyperbolic tangent function is -1 < x < 1, and its range is all real numbers.
02

Analyzing the given derivative formula

The formula provided is the derivative of the inverse hyperbolic tangent function, which is given by \(\frac{d}{dx}(\tanh^{-1}(x)) = \dfrac{1}{x^2 - 1}\). This formula gives the slope of the tangent line to the curve of the inverse hyperbolic tangent function at any point x in its domain.
03

Finding the valid interval for the formula

To determine the interval at which the given formula is valid, we need to find the domain of the function \(\dfrac{1}{x^2 - 1}\). This function is undefined when the denominator is equal to zero: \(x^2 - 1 = 0\) Solving for x, we get \(x=\pm1\). Therefore, the function is undefined when x = 1 and x = -1. Since the domain of the inverse hyperbolic tangent function is -1 < x < 1, and the given derivative formula is undefined at x = 卤1, it follows that the valid interval for the given formula is: -1 < x < 1

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