# In this article, we will explore the step-by-step process of finding the derivative of the function (x² + 1) / (2x + 1). We will use differentiation rules and techniques to simplify the expression and determine its derivative. Calculus, a branch of mathematics, is concerned with the study of rates of change and accumulation of quantities. Differentiation, a fundamental concept in calculus, is the process of finding the derivative of a function, which measures the rate at which the function changes with respect to its input.

## Understanding the Problem

Before we begin finding the derivative, let's closely examine the given function:

• f(x) = (x² + 1) / (2x + 1)

To find the derivative of this function, we will utilize differentiation rules and techniques, specifically the quotient rule. 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

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

In our case, g(x) = x² + 1, and h(x) = 2x + 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) = x² + 1
• g'(x) = 2x

Now, let's find h'(x) by differentiating h(x) with respect to x:

• h(x) = 2x + 1
• h'(x) = 2

### 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) = [(2x) * (2x + 1) - (x² + 1) * (2)] / (2x + 1)²

### Step 4: Simplifying the Expression

Now, let's simplify the expression further. We can expand and collect like terms in the numerator:

• f'(x) = [4x² + 2x - 2x² - 2] / (2x + 1)²

Combine like terms in the numerator:

• f'(x) = [2x² + 2x - 2] / (2x + 1)²

### Step 5: Final Simplification

The derivative can be further simplified by factoring out a common factor from the numerator:

• f'(x) = 2(x² + x - 1) / (2x + 1)²

Now, we have successfully found the derivative of the function (x² + 1) / (2x + 1):

• f'(x) = 2(x² + x - 1) / (2x + 1)²

## Conclusion

In this comprehensive guide, we have walked through the process of finding the derivative of the function (x² + 1) / (2x + 1) step by step. By applying the quotient rule and simplification techniques, we have obtained the final result:

• f'(x) = 2(x² + x - 1) / (2x + 1)²

Calculus and differentiation are fundamental concepts in mathematics and play a vital role in various fields such as physics, engineering, economics, and more. Understanding how to find derivatives is essential for analyzing functions and their behavior, and it forms the basis for solving a wide range of real-world problems.