# In this article, we will take you through the step-by-step process of finding the derivative of the function (2x² + 1) / (x² + 3x + 2). We will employ differentiation rules and techniques to simplify the expression and compute its derivative. Calculus is a branch of mathematics that deals with the concepts of change and motion. Differentiation, a fundamental operation in calculus, allows us to find the rate of change of a function with respect to its input variable.

## Understanding the Problem

Before we embark on finding the derivative, let's carefully examine the given function:

• f(x) = (2x² + 1) / (x² + 3x + 2)

To find the derivative of this function, we will apply the quotient rule, which is particularly useful for functions in the form u(x) / v(x), where both u(x) and v(x) are differentiable. The quotient rule is given by:

• 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) = 2x² + 1, and v(x) = x² + 3x + 2. To use the quotient rule, we must find the derivatives of both these functions.

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

Let's begin by finding u'(x), which is the derivative of u(x) with respect to x:

• u(x) = 2x² + 1
• u'(x) = 4x

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

• v(x) = x² + 3x + 2
• v'(x) = 2x + 3

### 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) = [(4x) * (x² + 3x + 2) - (2x² + 1) * (2x + 3)] / (x² + 3x + 2)²

### Step 4: Simplifying the Expression

Let's simplify the expression further by expanding and collecting like terms in the numerator:

• f'(x) = [4x³ + 12x² + 8x - 4x³ - 6x² - 2x - 3] / (x² + 3x + 2)²

Combine like terms in the numerator:

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

### Step 5: Factoring the Numerator

The derivative can be further simplified by factoring the numerator:

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

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

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

## Conclusion

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

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

Calculus and differentiation are fundamental concepts in mathematics with widespread 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 variety of real-world problems.