# In this article, we will delve into the process of finding the derivative of a product of two functions. Specifically, we will walk through the steps to find the derivative of the function: f(x) = (3x² - 2)(4x² + 1). Calculus, a branch of mathematics that explores rates of change and accumulation, plays a fundamental role in various scientific, engineering, and mathematical disciplines. A core concept in calculus is finding the derivative of a function, which provides valuable insights into how the function changes concerning its independent variable.

### Understanding the Product Rule

The product rule states that the derivative of the product of two functions u(x) and v(x) is given by:
• (uv)' = u'v + uv', Where u' and v' represent the derivatives of u(x) and v(x), respectively.
Now, let's apply the product rule to find the derivative of the given function, f(x) = (3x² - 2)(4x² + 1).

#### Step 1: Identify the Two Functions

In our case, u(x) = 3x² - 2 and v(x) = 4x² + 1.

#### Step 2: Find the Derivatives of the Two Functions

Now, let's find the derivatives of u(x) and v(x).
• Derivative of u(x): To find the derivative of u(x) = 3x² - 2 with respect to x, we'll apply the power rule: u'(x) = 6x + 0 = 6x.
• Derivative of v(x): To find the derivative of v(x) = 4x² + 1 with respect to x, we apply the power rule again: v'(x) = 8x + 0 = 8x.

#### Step 3: Apply the Product Rule

Now that we have found the derivatives of both functions, we can apply the product rule to find the derivative of their product, f(x):
• f'(x) = (u'v) + (uv').
Using the product rule, we get:
• f'(x) = [(6x)(4x² + 1)] + [(3x² - 2)(8x)].

#### Step 4: Simplify the Derivative

To simplify the derivative further, we will distribute and combine like terms:
• f'(x) = (24x³ + 6x) + (24x³ - 16x).
Now, combine like terms:
• f'(x) = 24x³ + 6x + 24x³ - 16x.
Combine like terms once more:
• f'(x) = 48x³ - 10x.
Thus the derivative of (3x² - 2)(4x² + 1) is 48x³ - 10x.

## Conclusion

In calculus, finding the derivative of a function is a fundamental skill that provides valuable information about how the function changes concerning its independent variable. In this article, we used the product rule, a key differentiation technique, to find the derivative of the function f(x) = (3x² - 2)(4x² + 1). The derivative we obtained is f'(x) = 48x³ - 10x. This derivative represents the rate of change of the original function and is essential for various applications in mathematics, science, and engineering. Understanding and applying differentiation rules like the product rule are crucial for analyzing complex relationships between variables and modeling real-world phenomena.