Problem 8 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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 = ±1, it follows that the valid interval for the given formula is: -1 < x < 1