# In this article, we will explore the step-by-step process of finding the derivative of the function (x² + 1) / (2x + 3). We will apply differentiation rules and techniques to simplify the expression and determine its derivative. Calculus, a branch of mathematics, is a powerful tool for analyzing functions and understanding how they change. One of the core concepts in calculus is differentiation, which allows us to calculate the rate of change of a function.

## Understanding the Problem

Before we begin, let's take a closer look at the given function:

• f(x) = (x² + 1) / (2x + 3)

To find the derivative of this function, we will utilize the quotient rule, a differentiation technique suitable for functions of the form u(x) / v(x), where both u(x) and v(x) are differentiable. The quotient rule states that:

• f'(x) = [u'(x) * v(x) - u(x) * v'(x)] / [v(x)]²

### Step 1: Identifying u(x) and v(x)

In our case, u(x) = x² + 1, and v(x) = 2x + 3. We will need to find the derivatives of both these functions in order to apply the quotient rule.

### Step 2: Finding u'(x) and v'(x)

To find u'(x), we'll differentiate u(x) with respect to x:

• u(x) = x² + 1
• u'(x) = 2x

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

• v(x) = 2x + 3
• v'(x) = 2

### Step 3: Applying the Quotient Rule

Now that we have u'(x), v'(x), u(x), and v(x), we can apply the quotient rule to find f'(x):

• f'(x) = [u'(x) * v(x) - u(x) * v'(x)] / [v(x)]²

• f'(x) = [(2x) * (2x + 3) - (x² + 1) * (2)] / (2x + 3)²

### Step 4: Simplifying the Expression

Now, let's simplify the expression further by expanding and collecting like terms in the numerator:

• f'(x) = [4x² + 6x - 2x² - 2] / (2x + 3)²

Combine like terms in the numerator:

• f'(x) = [2x² + 6x - 2] / (2x + 3)²

### Step 5: Final Simplification

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

• f'(x) = 2(x² + 3x - 1) / (2x + 3)²

Now, we have successfully found the derivative of the function (x² + 1) / (2x + 3):

• f'(x) = 2(x² + 3x - 1) / (2x + 3)²

## Conclusion

In this comprehensive guide, we have walked through the process of finding the derivative of the function (x² + 1) / (2x + 3) step by step. By applying the quotient rule and simplification techniques, we have obtained the final result:

• f'(x) = 2(x² + 3x - 1) / (2x + 3)²

Calculus and differentiation are fundamental concepts in mathematics and have wide-ranging applications 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.