D Dx X 1 X 2

A Deep Dive into the Derivative of x/(1+x²)

The expression d/dx [x/(1+x²)] represents the derivative of the function f(x) = x/(1+x²) with respect to x. This seemingly simple function underlies several important concepts in calculus and has applications in various fields. This article will explore the derivation of this derivative, its properties, and its significance in different contexts. We'll delve into various methods of solving this, discuss its graphical representation, and explore its applications.

Method 1: Quotient Rule

The most straightforward approach to finding the derivative of x/(1+x²) is using the quotient rule. The quotient rule states that the derivative of a function f(x) = g(x)/h(x) is given by:

f'(x) = [h(x)g'(x) - g(x)h'(x)] / [h(x)]²

In our case, g(x) = x and h(x) = 1 + x². Therefore, g'(x) = 1 and h'(x) = 2x. Applying the quotient rule:

f'(x) = [(1+x²)(1) - x(2x)] / (1+x²)²

Simplifying the expression:

f'(x) = (1 + x² - 2x²) / (1+x²)²

f'(x) = (1 - x²) / (1+x²)²

This is the derivative of x/(1+x²) using the quotient rule.

Method 2: Product Rule (with a twist)

While the quotient rule is the most direct method, we can also solve this using the product rule by rewriting the function:

f(x) = x * (1+x²)^(-1)

Now we can apply the product rule: f'(x) = g'(x)h(x) + g(x)h'(x), where g(x) = x and h(x) = (1+x²)^(-1).

First, we need to find the derivative of h(x) using the chain rule:

h'(x) = -1(1+x²)^(-2) * 2x = -2x/(1+x²)²

Now, applying the product rule:

f'(x) = 1*(1+x²)^(-1) + x*[-2x/(1+x²)²]

f'(x) = 1/(1+x²) - 2x²/(1+x²)²

To combine these terms, we need a common denominator:

f'(x) = [(1+x²) - 2x²] / (1+x²)²

f'(x) = (1 - x²) / (1+x²)²

As expected, we arrive at the same result as with the quotient rule.

Analyzing the Derivative: (1 - x²) / (1+x²)²

The derivative, (1 - x²) / (1+x²)², provides valuable information about the original function, x/(1+x²). Let's analyze its behavior:

• Critical Points: The derivative equals zero when the numerator is zero: 1 - x² = 0, which implies x = ±1. These are the critical points of the original function.

• Positive and Negative Values: The derivative is positive when 1 - x² > 0, which means -1 < x < 1. This indicates that the original function is increasing in this interval. Conversely, the derivative is negative when x < -1 or x > 1, implying the function is decreasing in these intervals.

• Inflection Points: To find potential inflection points, we would need to take the second derivative and set it to zero. This involves applying the quotient rule (or product rule) again to (1 - x²) / (1+x²)², which is a more complex calculation but will reveal points of concavity change.

• Asymptotic Behavior: As x approaches positive or negative infinity, the function x/(1+x²) approaches zero. This is because the denominator grows much faster than the numerator. The derivative also approaches zero as x tends towards infinity, confirming this asymptotic behavior.

Graphical Representation

Plotting the original function, f(x) = x/(1+x²), and its derivative, f'(x) = (1 - x²) / (1+x²)², reveals a clear relationship:

• The original function shows a bell-shaped curve, symmetric around the y-axis, with a maximum at x = 1 and a minimum at x = -1.

• The derivative is positive where the original function is increasing, and negative where it's decreasing. The derivative is zero at the maximum and minimum points of the original function.

This visual confirmation reinforces our analytical findings.

Applications

The function x/(1+x²) and its derivative appear in several areas of mathematics and its applications:

• Probability and Statistics: This function resembles the probability density function (PDF) of certain probability distributions, albeit with appropriate scaling factors. Its derivative helps analyze properties like the mean and variance of those distributions.

• Signal Processing: The function can model various signal characteristics. The derivative helps in analyzing signal changes, identifying peaks and troughs, and designing filters.

• Physics: This function, or similar forms, can appear in modeling physical phenomena where damped oscillations or resonant behavior are involved. Its derivative helps analyze the rate of change of these phenomena.

• Computer Graphics: Functions of this form are used in creating smooth curves and surfaces. The derivative is crucial in determining tangent lines and normals, which are essential for rendering algorithms.

Further Exploration: Second Derivative

The second derivative helps determine the concavity of the original function. Applying the quotient rule (or product rule) to f'(x) = (1 - x²) / (1+x²)² results in a rather complex expression. This second derivative, when set equal to zero, identifies inflection points – points where the concavity of the graph changes. Solving for these points provides a more complete understanding of the function's behavior.

Conclusion

The seemingly simple expression d/dx [x/(1+x²)] leads to a rich exploration of calculus concepts, including the quotient rule, product rule, chain rule, critical points, increasing and decreasing intervals, and asymptotic behavior. Understanding its derivative is not just about finding a formula; it's about gaining insights into the properties and applications of the original function in various fields. The analysis performed here provides a strong foundation for further investigation into more complex functions and their derivatives. By understanding this example, one can better grasp the power and versatility of calculus in solving real-world problems. The detailed analysis offered here illustrates how to approach similar problems, encouraging a deeper understanding of differential calculus. Remember that practice and the application of the rules are key to mastering these concepts.