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A derivative is a fundamental concept in calculus that represents the rate at which a function is changing at any given point. More formally, it's the slope of the tangent line to the curve of the function at a point. For the function given, \[ f(x) = \tan^{-1}(x^2 - 1) \]we begin by finding its first derivative using the chain rule.
The chain rule allows us to take the derivative of a composite function by differentiating its outer and inner parts separately. Here, the outer function is \( \tan^{-1}(u) \) and the inner function is \( u = x^2 - 1 \).

• Differentiate the outer function: \( \frac{d}{du} \tan^{-1}(u) = \frac{1}{1+u^2} \)
• Differentiate the inner function: \( \frac{d}{dx}(x^2-1) = 2x \)

Multiply them together to get the first derivative of the entire function: \[ f'(x) = \frac{2x}{1 + (x^2 - 1)^2} \] This result helps us determine the behavior of the function over different intervals.

In calculus, concavity describes the direction in which a curve opens. A function is concave up if it opens upwards like a cup, and concave down if it opens downwards. We determine the concavity of a function by using its second derivative.
For our function \( f(x) = \tan^{-1}(x^2 - 1) \), we take the second derivative using the quotient rule. The second derivative is: \[ f''(x) = \frac{2(1 + (x^2-1)^2) - 8x^4 + 8x^2}{(1 + (x^2 - 1)^2)^2} \] By evaluating this, we determine intervals where \( f''(x) > 0 \) and \( f''(x) < 0 \).

• If \( f''(x) > 0 \), the function is concave up on that interval.
• If \( f''(x) < 0 \), the function is concave down.

This analysis shows that our function is concave up in the interval \((1, \infty)\) and concave down in \((-\infty, 1)\). These intervals tell us about the "bending" direction of the curve over its domain.

Inflection points are key locations where a function's concavity changes from upwards to downwards or vice versa. To find these points, we look for values of \( x \) at which the second derivative changes sign. In other words, inflection points occur where \( f''(x) = 0 \) or is undefined, and the sign of \( f''(x) \) actually changes.
For the given function, our analysis of the second derivative reveals that the concavity changes at \( x = 1 \). This point marks a transition from concave down to concave up, indicating an inflection point. Additionally, due to symmetrical properties of the function's curve, a change is also noted at \( x = -1 \).

• An inflection point at \( x = 1 \) means the graph changes concavity here.
• The function's symmetry suggests another inflection point at \( x = -1 \).

These specific \( x \)-coordinates highlight the smooth shift in the "bending" nature of the function.

Increasing and decreasing intervals of a function are identified by the behavior of its first derivative. A function is increasing on intervals where its derivative is positive and decreasing where its derivative is negative.
For the given function \( f(x) = \tan^{-1}(x^2 - 1) \), we analyze its first derivative \[ f'(x) = \frac{2x}{1 + (x^2 - 1)^2} \]to find these intervals.

• \( f'(x) > 0 \) implies that the function is increasing, which happens for \( x > 0 \).
• \( f'(x) < 0 \) implies the function is decreasing, which is the case for \( x < 0 \).

This tells us that the function increases on the interval \((0, \infty)\) and decreases on \((-\infty, 0)\). These intervals provide insights into how the function is trending across its domain, helping to sketch or understand its graph better.