How to find derivative of (x) / (x² + 1) ?

In this article, we will explore the step-by-step process of finding the derivative of the function (x) / (x² + 1). We will use differentiation rules and techniques to simplify the expression and find its derivative. Calculus is a branch of mathematics that deals with rates of change and is an essential tool in various fields such as physics, engineering, economics, and more. One fundamental concept in calculus is finding derivatives, which allows us to understand how a function changes as its input varies. 

Understanding the Problem

Before we begin finding the derivative, let's take a closer look at the given function:
  • f(x) = (x) / (x² + 1)
To find the derivative of this function, we'll employ several differentiation techniques, including the quotient rule and simplification. The quotient rule states that if you have a function of the form g(x) / h(x), where both g(x) and h(x) are differentiable, then the derivative can be found using the formula:
  • f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]²
Now, let's break down the problem step by step.

Step 1: Identifying g(x) and h(x)

In our case, g(x) = x, and h(x) = x² + 1. We will need to find the derivatives of both these functions to apply the quotient rule.

Step 2: Finding g'(x) and h'(x)

To find g'(x), we'll differentiate g(x) with respect to x:
  • g(x) = x
  • g'(x) = 1
Now, let's find h'(x) by differentiating h(x) with respect to x:
  • h(x) = x² + 1
  • h'(x) = 2x

Step 3: Applying the Quotient Rule

Now that we have g'(x), h'(x), g(x), and h(x), we can apply the quotient rule to find f'(x):
  • f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]²
  • f'(x) = [(1) * (x² + 1) - (x) * (2x)] / (x² + 1)²

Step 4: Simplifying the Expression

Now, we need to simplify the expression further. To do this, we can start by expanding and collecting like terms in the numerator:
  • f'(x) = [x² + 1 - 2x²] / (x² + 1)²
Notice that the x² terms cancel out, leaving us with:
  • f'(x) = [1 - x²] / (x² + 1)²

Step 5: Final Simplification

The derivative can be further simplified by factoring out a common factor from the numerator:
  • f'(x) = (1 - x²) / (x² + 1)²
Now, we have successfully found the derivative of the function (x) / (x² + 1):
  • f'(x) = (1 - x²) / (x² + 1)²

Thus the derivative of (x) / (x² + 1) ? is (1 - x²) / (x² + 1)²

Conclusion

In this comprehensive guide, we walked through the process of finding the derivative of the function (x) / (x² + 1) step by step. By using the quotient rule and simplification techniques, we arrived at the final result:
  • f'(x) = (1 - x²) / (x² + 1)²
Derivatives are a crucial concept in calculus and are used to analyze various aspects of functions and their behavior. Understanding how to find derivatives is fundamental in mathematics and is applicable in solving real-world problems in various fields.

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