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Calculating the derivative of a function helps us understand how the function's value changes with respect to changes in the input. In the exercise, we were tasked with finding the derivative of the function \( f(x) = \sqrt{x^2 - 1} \). The derivative gives us the function's rate of change or its slope.

To find the derivative, we used the chain rule, a fundamental technique in calculus. The chain rule is especially useful when we deal with composite functions. Here is how it works:

• Start by differentiating the outer function (in this case, the square root) leaving the inner function intact.
• Differentiating \( \sqrt{u} \) gives us \( \frac{1}{2\sqrt{u}} \).
• This differentiation results in \( \frac{1}{2\sqrt{x^2 - 1}} \) initially.
• Next, differentiate the inner function \( x^2 - 1 \) with respect to \( x \), resulting in \( 2x \).
• The derivative of the original function by the chain rule is then the product of these, which simplifies to \( \frac{x}{\sqrt{x^2 - 1}} \).

Calculating derivatives accurately, as shown here, is critical for later steps in analyzing the function's behavior.

Critical points of a function occur where the derivative is zero or undefined, and they tell us places where the function's slope changes from increasing to decreasing or vice versa. However, in this instance, looking at our derivative \( f'(x) = \frac{x}{\sqrt{x^2 - 1}} \), we see that it never equals zero for real values of \( x \).

But, critical points also include where the derivative does not exist, which happens when the denominator is zero or negative. Here, the denominator, \( \sqrt{x^2 - 1} \), is undefined when \( |x| < 1 \) because the expression inside the square root becomes negative.

Thus, we identify \( x = 1 \) and \( x = -1 \) as borders of undefined behavior for the derivative. These points mark the boundaries of the function's domain, guiding us in determining where to inspect for increases and decreases.

Determining where a function is increasing or decreasing provides a picture of how the function behaves overall. By checking the sign of the derivative, we can track these intervals:

• The sign of the derivative \( f'(x) = \frac{x}{\sqrt{x^2-1}} \) tells us when the function is increasing or decreasing.
• For \( x > 1 \), \( f'(x) \) is positive, indicating that the function is increasing on the interval \((1, \infty)\).
• For \( x < -1 \), \( f'(x) \) is negative, suggesting the function is decreasing over the interval \((-\infty, -1)\).
• The function is not defined between \(-1 < x < 1\), as the original expression under the square root becomes negative.
• There are no constant intervals because the derivative never equals zero.

This analysis guides us in understanding the overall shape and movement of the function on different intervals.