Derivative Of (3x³ - 2x² + 1)^(5/2): Step-by-Step Guide

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Hey guys! Today, we're diving into a fun calculus problem: finding the derivative of (3x³ - 2x² + 1)^(5/2) with respect to x. This might look intimidating at first, but don't worry! We'll break it down step by step, making it super easy to understand. Grab your pencils, and let's get started!

Understanding the Problem

Before we jump into the solution, let’s make sure we understand exactly what we're trying to do. The derivative of a function tells us how the function changes as its input changes. In simpler terms, it gives us the slope of the tangent line at any point on the function's graph. For our function, (3x³ - 2x² + 1)^(5/2), we want to find a new function that tells us this slope for any value of x.

Why is this important? Derivatives are used everywhere in science and engineering. They help us understand rates of change, optimize processes, and model all sorts of real-world phenomena. So, mastering derivatives is a crucial step in your calculus journey. This particular problem involves a composite function (a function inside another function) and a power, which means we'll be using the chain rule and the power rule. These are fundamental concepts in differentiation, and this problem is a fantastic way to practice them. When you encounter a problem like (3x³ - 2x² + 1)^(5/2), recognize that it's a composition of functions. The outer function is something raised to the power of 5/2, and the inner function is the polynomial 3x³ - 2x² + 1. This recognition is key to applying the chain rule correctly. Think of the chain rule as peeling an onion, layer by layer. You differentiate the outermost layer first, then move inward, differentiating each subsequent layer and multiplying the results together. It might seem complex now, but with practice, it will become second nature.

Tools We'll Use: Chain Rule and Power Rule

To solve this, we'll primarily use two important rules from calculus:

  1. Chain Rule: This rule helps us differentiate composite functions (functions within functions). If we have a function y = f(g(x)), the chain rule states that dy/dx = f'(g(x)) * g'(x).
  2. Power Rule: This rule is used to differentiate powers of x. If we have a function y = x^n, the power rule states that dy/dx = n * x^(n-1).

These two rules are the bread and butter of differentiation, especially when dealing with more complex functions. Let’s understand them a bit better.

Chain Rule Explained

The chain rule is essential when you have a function nested inside another function. Imagine you have a machine that depends on another machine. The chain rule helps you understand how changes in the input of the first machine affect the output of the entire system. In mathematical terms, if y is a function of u, and u is a function of x, then the rate of change of y with respect to x is the product of the rate of change of y with respect to u and the rate of change of u with respect to x. This can be written as dy/dx = (dy/du) * (du/dx). The chain rule allows us to break down a complicated derivative into smaller, more manageable parts. By identifying the inner and outer functions, we can differentiate each separately and then multiply the results together to get the overall derivative. Understanding the chain rule is not just about memorizing a formula; it's about understanding how different parts of a function interact and affect each other's rates of change.

Power Rule Explained

The power rule is a straightforward way to differentiate terms that involve x raised to a power. It simply states that if you have a term like x^n, its derivative is n * x^(n-1). This means you multiply the term by the exponent and then reduce the exponent by 1. For example, the derivative of x³ is 3x², and the derivative of x^(5/2) is (5/2)x^(3/2). The power rule is a fundamental tool in calculus and is used extensively in many different types of differentiation problems. It is derived from the definition of the derivative and is applicable for any real number n, whether it's an integer, a fraction, or even a negative number. Knowing the power rule by heart will save you time and effort in many calculus problems.

Step-by-Step Solution

Okay, let's apply these rules to our problem:

Function: y = (3x³ - 2x² + 1)^(5/2)

  1. Apply the Chain Rule: First, we treat the entire expression inside the parentheses as a single entity. Let u = 3x³ - 2x² + 1. Then, y = u^(5/2). So, dy/du = (5/2) * u^(3/2).

  2. Differentiate the Inside Function: Now, we need to find du/dx. Remember, u = 3x³ - 2x² + 1. Differentiating this with respect to x, we get du/dx = 9x² - 4x.

  3. Combine the Results: Using the chain rule, dy/dx = (dy/du) * (du/dx) = (5/2) * u^(3/2) * (9x² - 4x).

  4. Substitute Back: Replace u with 3x³ - 2x² + 1. So, dy/dx = (5/2) * (3x³ - 2x² + 1)^(3/2) * (9x² - 4x).

  5. Simplify: We can simplify this expression a bit by factoring out an x from the second term: dy/dx = (5/2) * x * (9x - 4) * (3x³ - 2x² + 1)^(3/2).

And that’s it! We've found the derivative.

Let's Break it Down Further

Step 1: Recognizing the Composite Function

The very first step in solving this problem is recognizing that you're dealing with a composite function. This means that you have a function inside another function. In our case, the outer function is raising something to the power of 5/2, and the inner function is the polynomial 3x³ - 2x² + 1. Recognizing this structure is key to applying the chain rule correctly. If you try to differentiate this function without recognizing the composite nature, you'll likely make a mistake. Always look for nested functions first! Identifying the outer and inner functions will guide you on how to apply the chain rule step by step. This initial recognition makes the rest of the problem much more manageable.

Step 2: Applying the Chain Rule

The chain rule is the heart of this problem. Once you've identified the outer and inner functions, you need to differentiate each separately and then multiply the results together. In our case, we first differentiated the outer function (something raised to the power of 5/2) with respect to the inner function. This gave us (5/2) * (3x³ - 2x² + 1)^(3/2). Then, we differentiated the inner function (3x³ - 2x² + 1) with respect to x, which gave us 9x² - 4x. Finally, we multiplied these two results together to get the overall derivative. The chain rule ensures that you account for the rate of change of both the inner and outer functions, giving you the correct derivative of the composite function. Practice applying the chain rule to various composite functions to become more comfortable with this process.

Step 3: Differentiating the Inner Function

Differentiating the inner function, 3x³ - 2x² + 1, involves applying the power rule to each term. Remember, the power rule states that the derivative of x^n is n * x^(n-1). So, the derivative of 3x³ is 9x², and the derivative of -2x² is -4x. The derivative of the constant 1 is 0. Combining these results, we get 9x² - 4x. This step is a straightforward application of the power rule, but it's crucial to get it right to ensure the overall derivative is correct. Double-check your work to avoid common mistakes, such as forgetting to multiply by the exponent or decreasing the exponent by 1.

Step 4: Combining and Simplifying

After applying the chain rule and differentiating both the outer and inner functions, the final step is to combine the results and simplify the expression. This involves multiplying the derivative of the outer function by the derivative of the inner function. In our case, this gave us (5/2) * (3x³ - 2x² + 1)^(3/2) * (9x² - 4x). We then simplified this expression by factoring out an x from the second term, resulting in (5/2) * x * (9x - 4) * (3x³ - 2x² + 1)^(3/2). Simplifying the expression makes it easier to read and work with in further calculations. Always look for opportunities to simplify your derivative to make it as clean as possible.

Final Answer

So, the derivative of (3x³ - 2x² + 1)^(5/2) with respect to x is:

(5/2) * x * (9x - 4) * (3x³ - 2x² + 1)^(3/2)

Conclusion

Great job, guys! We've successfully found the derivative of a complex function using the chain rule and power rule. Remember, practice is key to mastering these concepts. Keep solving problems, and you'll become a differentiation pro in no time!

If you have any questions or want to explore more calculus problems, feel free to ask. Happy differentiating!