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The first derivative is an excellent tool for understanding the behavior of a function. It helps determine when a graph is increasing or decreasing. For the function \( y = \tanh^{-1} x \), we need the derivative to analyze its growth pattern. The derivative of inverse hyperbolic functions works slightly differently than regular trigonometric functions. Here, the derivative of the function \( y = \tanh^{-1} x \) is \( \frac{1}{1-x^2} \).
To determine if the function is increasing, we look at the sign of this derivative. An increasing function will have a derivative greater than zero. In this case, \( \frac{1}{1-x^2} > 0 \) as long as the denominator, \( 1 - x^2 \), is positive.

• The term \( 1-x^2 \) is positive for \( -1 < x < 1 \), meaning the fraction \( \frac{1}{1-x^2} \) is also positive in this interval.
• Thus, the function \( y = \tanh^{-1} x \) is always increasing for these values of \( x \).

The second derivative provides insights into the concavity of a function and helps us identify inflection points. For the function \( y = \tanh^{-1} x \), the second derivative is \( \frac{2x}{(1-x^2)^2} \). This derivative represents the rate of change of the first derivative.
An inflection point is where the function changes its concavity, which is indicated by a sign change in the second derivative. To identify such points for our function, we need:

• To find when the second derivative equals zero: \( \frac{2x}{(1-x^2)^2} = 0 \) simplifies to \( x = 0 \).
• To confirm a change in sign around \( x = 0 \), which verifies an inflection point. As \( x \) moves through zero, from negative to positive, \( \frac{2x}{(1-x^2)^2} \) shifts from negative to positive.

Therefore, the origin \( (0,0) \) is an inflection point for our function.

Hyperbolic functions are similar to their trigonometric counterparts but are based on hyperbolas instead of circles. They play a crucial role in many areas of mathematics. The function \( \tanh^{-1} x \), known as the inverse hyperbolic tangent, is part of this family of functions. These functions are defined using exponential functions.

• The hyperbolic tangent function, \( \tanh x \), can be expressed as \( \frac{e^x - e^{-x}}{e^x + e^{-x}} \).
• Its inverse, \( y = \tanh^{-1} x \), is then used to undo the effects of \( \tanh x \).

Key hyperbolic functions have properties similar to the familiar sine and cosine functions, but they often describe real-world phenomena better, such as temperature distribution, catenary shapes, and electrical engineering models. Understanding the derivatives of these functions, as we did with \( y = \tanh^{-1} x \), is essential for analyzing their behavior.