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The derivative is a crucial concept in calculus that helps us understand how a function changes. For a function like \( f(x) = \frac{\sin x}{1+\cos^{2}x} \), we use the derivative to find the different features of the function, such as where it increases or decreases. To find the derivative of this function, we use the quotient rule:

• Quotient Rule: If you have a function in the form of \( \frac{u}{v} \), the derivative is \( \frac{{v\cdot u' - u\cdot v'}}{v^2} \)

Applying the quotient rule to the function, we find the derivative to be:\[f'(x) = \frac{(1+\cos^{2}x) \cdot \cos x - \sin x \cdot (-2\cos x\sin x)}{(1+\cos^{2}x)^{2}}\]Simplifying further, this becomes:\[f'(x) = \frac{\cos x + \cos^{3}x + 2\cos^{2}x\sin x}{(1+\cos^{2}x)^{2}}\]This derivative gives us the rate of change of the function at any given point.

Critical points of a function occur where the first derivative is zero or undefined. These points are important as they may indicate potential relative extrema (maximums or minimums).For the function \(f(x) = \frac{\sin x}{1+\cos^{2}x}\), we identify critical points by setting the derivative \(f'(x)\) equal to zero:\[\frac{\cos x + \frac{1}{2}\cos^{3} x }{(1+\cos^{2}x)^{2}} = 0\]This results in \(\cos x = 0\) or \(\cos^{2}x = \frac{-1}{2}\). However, since \(\cos^{2}x = \frac{-1}{2}\) is impossible with real values, our critical points are where \(\cos x = 0\).We find these points at:

• \(x = \frac{\pi}{2}\)
• \(x = \frac{3\pi}{2}\)

These points are our candidates for relative extrema.

Relative extrema refer to the points on a function's graph where the function reaches a local maximum or minimum. We apply the First Derivative Test to determine whether each critical point is a relative maximum, minimum, or neither.Based on the previously identified critical points:

• At \(x = \frac{\pi}{2}\), the derivative \(f'(x)\) changes from positive to negative, indicating a transition from increasing to decreasing. Thus, this point is a relative maximum.
• At \(x = \frac{3\pi}{2}\), the derivative \(f'(x)\) changes from negative to positive, which shows the function moves from decreasing to increasing. Therefore, this point is a relative minimum.

This test is a powerful tool to quickly identify peaks (maximum) and valleys (minimum) on a graph.

To identify intervals where a function increases or decreases, we look at the sign of the first derivative \(f'(x)\). Here's how:

• If \(f'(x) > 0\), the function is increasing in that interval.
• If \(f'(x) < 0\), the function is decreasing in that interval.

Using our calculated derivative for the function \(f(x)\), we create a number line test through the critical points \(x = \frac{\pi}{2}\) and \(x = \frac{3\pi}{2}\):

• For \((0, \frac{\pi}{2})\), \(f'(x) > 0\), indicating the function is increasing.
• For \((\frac{\pi}{2}, \frac{3\pi}{2})\), \(f'(x) < 0\), indicating the function is decreasing.
• For \((\frac{3\pi}{2}, 2\pi)\), \(f'(x) > 0\), indicating the function is again increasing.

This analysis divides the domain into intervals, providing a clearer picture of the function's behavior.