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The inverse tangent function, denoted as \( \tan^{-1} y \) or \( \arctan y \), is the inverse of the tangent function. It provides the angle whose tangent is the given number \( y \). When you see \( \tan^{-1} y \), essentially you are asking, "What angle has a tangent of \( y \)?" This function is crucial when dealing with angles and trigonometry-related problems in calculus.

The range of the inverse tangent function is restricted to \((-\frac{\pi}{2}, \frac{\pi}{2})\), ensuring that it is a true function with each input providing a unique output.
Understanding how to differentiate functions like \( \tan^{-1} y \) is essential in calculus, especially when determining the rate of change of angles in relation to other variables.

Differentiation is a core concept in calculus that deals with finding the rate at which a function changes. In our exercise, we start by differentiating the inverse tangent function: \( u = \tan^{-1} y \). Applying the known derivative formula, the first derivative is \( \frac{du}{dy} = \frac{1}{1+y^2} \).

This result tells us how \( u \) changes with respect to \( y \). To get the second derivative, which is \( \frac{d^2u}{dy^2} \), we must differentiate \( \frac{1}{1+y^2} \) again with respect to \( y \). This process is called taking higher-order derivatives and helps to understand the curvature and concavity of the function related to its graph.

• The first derivative indicates the slope or the rate of change.
• The second derivative provides information about acceleration or how the rate of change itself changes.

Recognizing the importance of differentiation allows us to analyze and predict behavior in various mathematical models and real-world scenarios.

The quotient rule is a method used when differentiating ratios of two functions, say \( \frac{f(y)}{g(y)} \). It is given by the formula: \[ \frac{d}{dy} \left( \frac{f(y)}{g(y)} \right) = \frac{f'(y)g(y) - f(y)g'(y)}{[g(y)]^2} \] This helps systematically tackle differentiation when dealing with complicated fractions.

In our solution, we used the quotient rule to differentiate \( \frac{1}{1+y^2} \). By diagnosing \( f(y) = 1 \) and \( g(y) = 1 + y^2 \), we computed their derivatives: \( f'(y) = 0 \) and \( g'(y) = 2y \). The quotient rule guides us in combining these derivatives correctly to obtain the second derivative: \[ \frac{d^2u}{dy^2} = \frac{-2y}{(1+y^2)^2} \]
Using the quotient rule correctly ensures accuracy in more complex environments where simple rules of differentiation won't apply.