Differentiating $4x^2(x^3 + 1)^3$: A Step-by-Step Guide

Hey math enthusiasts! Today, we're going to dive into the exciting world of calculus and learn how to differentiate the function $4x^2(x^3 + 1)^3$. This might seem a little intimidating at first, but trust me, with a step-by-step approach, we'll break it down into manageable chunks. So, buckle up, grab your pens and paper, and let's get started! This derivation involves a combination of the product rule and the chain rule, two fundamental concepts in differential calculus. We'll walk through each step meticulously, ensuring you understand every aspect of the process. Our goal isn't just to get the answer but to understand why we're doing what we're doing. This deep understanding will help you tackle more complex problems in the future. We'll make sure that by the end of this guide, you'll be able to confidently differentiate similar functions. Ready? Let's go!

Understanding the Product Rule

Before we jump into the problem, let's refresh our memory on the product rule. The product rule is used to differentiate functions that are the product of two other functions. In simpler terms, if we have a function $f(x) = u(x)v(x)$, then its derivative, $f'(x)$, is given by:

$f'(x) = u'(x)v(x) + u(x)v'(x)$

Where:

• $u'(x)$ is the derivative of $u(x)$
• $v'(x)$ is the derivative of $v(x)$

Think of it like this: The derivative of the product is the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function. Makes sense, right? This rule is crucial for our problem because we have a product of two functions: $4x^2$ and $(x^3 + 1)^3$. So, the first key to solving this derivation is to understand and apply this rule. The product rule is one of the foundational stones in calculus, so a good grasp of it will make tackling all sorts of derivative problems easier. Practice makes perfect, so don't be afraid to try out more examples to solidify your understanding. It's all about practice and repetition, which builds confidence and ensures that you can apply it in a variety of situations. Once you're comfortable with the product rule, you're one step closer to mastering this derivative. Remember to keep the rule in mind as you move forward. We'll be using it extensively throughout this derivation. It is important to know that the order in the product rule does not matter, as addition is commutative.

Breaking Down the Function

Now, let's look at our function: $4x^2(x^3 + 1)^3$. We can see that it's a product of two terms. Let's define:

• $u(x) = 4x^2$
• $v(x) = (x^3 + 1)^3$

Our next step is to find the derivatives of $u(x)$ and $v(x)$. This will involve the power rule and the chain rule, respectively. So, let's get those derivatives!

The first part is $u(x) = 4x^2$. This one is pretty straightforward. Using the power rule (which states that the derivative of $x^n$ is $nx^{n-1}$), we get:

$u'(x) = 8x$

Easy peasy, right? Now, let's move on to the more complex part: $v(x) = (x^3 + 1)^3$. This requires the chain rule.

To make things easier, we've broken the original problem down into smaller chunks that are more manageable. This is a common strategy when solving complex calculus problems: divide and conquer. This way, the process becomes less intimidating and you can focus on one small part at a time. Breaking down the problem helps avoid mistakes and keeps the process organized. By defining the separate components, we can apply the rules in an organized manner. This also helps with error checking, as we can check each part's derivate to make sure it is correct, before going on to the next step. Remembering to break down the original problem is a useful tool to help with this and other calculus problems, so you will want to get comfortable using it. It might seem like an extra step, but breaking the function down will allow us to utilize both the product rule and chain rule more easily.

Applying the Chain Rule

The chain rule is used to differentiate composite functions, which are functions within functions. In our case, we have $(x^3 + 1)$ raised to the power of 3. The chain rule states:

If $y = f(g(x))$, then $y' = f'(g(x)) * g'(x)$

Let's apply this to $v(x) = (x^3 + 1)^3$. We can consider:

• $g(x) = x^3 + 1$
• $f(g) = g^3$

So, we need to find $g'(x)$ and $f'(g)$.

First, let's find $g'(x)$. The derivative of $x^3 + 1$ is:

$g'(x) = 3x^2$

Next, let's find $f'(g)$. The derivative of $g^3$ is:

$f'(g) = 3g^2$

Now, we can apply the chain rule:

$v'(x) = f'(g(x)) * g'(x) = 3(x^3 + 1)^2 * 3x^2 = 9x^2(x^3 + 1)^2$

Awesome! We now have both $u'(x)$ and $v'(x)$.

The chain rule might seem a little tricky at first, but with practice, it becomes second nature. It's used so frequently in calculus that mastering it is essential. Remember to identify the inner and outer functions carefully. In our case, the inner function is $x^3 + 1$, and the outer function is the power of 3. Keep in mind that when applying the chain rule, you differentiate the outer function first, and then multiply by the derivative of the inner function. This process of working from the outside in is important. Practicing with different types of functions and variations of the chain rule will help to make you confident with this concept. The more you apply the chain rule, the more comfortable you'll become, and the faster you'll be able to work through these problems.

Putting it All Together

Now that we have all the pieces, let's put them together using the product rule:

$f'(x) = u'(x)v(x) + u(x)v'(x)$

Substitute the values we found earlier:

$f'(x) = (8x)(x^3 + 1)^3 + (4x^2)(9x^2(x^3 + 1)^2)$

Now, let's simplify and factorize. This is where things get a bit cleaner:

$f'(x) = 8x(x^3 + 1)^3 + 36x^4(x^3 + 1)^2$

We can factor out $4x(x^3 + 1)^2$:

$f'(x) = 4x(x^3 + 1)^2[2(x^3 + 1) + 9x^2]$

Simplify the terms inside the brackets:

$f'(x) = 4x(x^3 + 1)^2(2x^3 + 2 + 9x^2)$

Finally, reorder terms:

$f'(x) = 4x(x^3 + 1)^2(2x^3 + 9x^2 + 2)$

And there you have it! The derivative of $4x^2(x^3 + 1)^3$ is $4x(x^3 + 1)^2(2x^3 + 9x^2 + 2)$. We have successfully used both the product rule and the chain rule to solve the problem. High five! We've made it through the entire derivation process, and we should be very proud of what we've accomplished. It might seem like a lot, but by breaking the function down, using the rules, and carefully applying each step, we have solved the problem. It is time to celebrate our success, but remember to keep practicing. This will help you to become more proficient in calculus and enable you to solve even more complex problems. With practice, you will become faster, and find the whole process easier to manage. Now you can use this understanding and technique on many more problems.

Final Answer

So, the derivative of $4x^2(x^3 + 1)^3$ is:

$f'(x) = 4x(x^3 + 1)^2(2x^3 + 9x^2 + 2)$

Great job, everyone! You've successfully navigated through the product rule, the chain rule, and the simplification steps to arrive at the final answer. Keep practicing, and you'll become a calculus whiz in no time. If you got stuck at any point, go back and review the sections that caused trouble. And remember, asking for help is always a good idea. Math can be challenging, but it's also incredibly rewarding. Keep up the excellent work, and enjoy the journey!

I hope this guide has been helpful. If you have any questions or want to try another example, feel free to ask. Happy differentiating!