Differentiating Y = (2x - 2)^5(1 - X^3)^4: A Step-by-Step Guide

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Hey guys! Today, we're diving into a calculus problem where we need to differentiate a somewhat complex function. Specifically, we're going to tackle the function y = (2x - 2)^5(1 - x3)4. This problem looks intimidating at first, but don't worry! We'll break it down step-by-step using the product rule and the chain rule. So, grab your pencils, and let's get started!

Understanding the Problem and Necessary Tools

Before we jump into the solution, let's make sure we understand what we're dealing with. We have a function, y, that is the product of two expressions: (2x - 2)^5 and (1 - x3)4. To differentiate this, we'll need two key tools from calculus: the product rule and the chain rule.

The Product Rule

The product rule is essential when you're differentiating a function that is the product of two other functions. If we have a function y = u(x)v(x), where u(x) and v(x) are both functions of x, then the derivative dy/dx is given by:

dy/dx = u'(x)v(x) + u(x)v'(x)

In simpler terms, the derivative of the product is the derivative of the first function times the second function, plus the first function times the derivative of the second function. Keep this formula handy; we'll be using it soon!

The Chain Rule

The chain rule comes into play when we're differentiating a composite function – a function within a function. If we have y = f(g(x)), then the derivative dy/dx is given by:

dy/dx = f'(g(x)) * g'(x)

Basically, you differentiate the outer function, keeping the inner function as is, and then multiply by the derivative of the inner function. Think of it like peeling an onion layer by layer. This rule will be crucial for differentiating the expressions (2x - 2)^5 and (1 - x3)4 because they are functions raised to a power.

Applying the Product Rule: Setting Up the Problem

Okay, now that we've refreshed our memory on the product and chain rules, let's apply them to our function, y = (2x - 2)^5(1 - x3)4. To make things clearer, let's identify our u(x) and v(x):

  • u(x) = (2x - 2)^5
  • v(x) = (1 - x3)4

Now, according to the product rule, we need to find u'(x) and v'(x). This is where the chain rule enters the picture!

Differentiating u(x) Using the Chain Rule

Let's find the derivative of u(x) = (2x - 2)^5. We'll apply the chain rule here. Think of (2x - 2) as our inner function and the power of 5 as our outer function.

  1. Differentiate the outer function: Bring down the power and reduce it by 1: 5(2x - 2)^4.
  2. Differentiate the inner function: The derivative of (2x - 2) is simply 2.
  3. Multiply the results: u'(x) = 5(2x - 2)^4 * 2 = 10(2x - 2)^4

So, we've found that u'(x) = 10(2x - 2)^4. Awesome! Now let's move on to finding v'(x).

Differentiating v(x) Using the Chain Rule

Next up, we need to find the derivative of v(x) = (1 - x3)4. We'll use the chain rule again. This time, our inner function is (1 - x^3) and the outer function is the power of 4.

  1. Differentiate the outer function: Bring down the power and reduce it by 1: 4(1 - x3)3.
  2. Differentiate the inner function: The derivative of (1 - x^3) is -3x^2.
  3. Multiply the results: v'(x) = 4(1 - x3)3 * (-3x^2) = -12x^2(1 - x3)3

Great! We've found that v'(x) = -12x^2(1 - x3)3. We now have all the pieces we need to apply the product rule.

Putting It All Together: Applying the Product Rule

Now that we have u(x), v(x), u'(x), and v'(x), we can finally plug them into the product rule formula:

dy/dx = u'(x)v(x) + u(x)v'(x)

Substituting our values, we get:

dy/dx = [10(2x - 2)^4](1 - x3)4 + (2x - 2)5[-12x2(1 - x3)3]

This looks pretty complicated, but we're not done yet! We can simplify this expression further.

Simplifying the Derivative: Factoring Out Common Terms

To simplify the derivative, we need to look for common factors in both terms of the expression. Notice that both terms have factors of (2x - 2)^4 and (1 - x3)3. Let's factor these out:

dy/dx = (2x - 2)^4(1 - x3)3 [10(1 - x^3) - 12x^2(2x - 2)]

Factoring out common terms makes the expression much cleaner and easier to work with. Now, let's simplify the expression inside the brackets.

Further Simplification: Expanding and Combining Like Terms

We need to expand the expression inside the brackets and combine like terms to simplify further:

dy/dx = (2x - 2)^4(1 - x3)3 [10 - 10x^3 - 24x^3 + 24x^2]

Combine the x^3 terms:

dy/dx = (2x - 2)^4(1 - x3)3 [10 - 34x^3 + 24x^2]

Now, let's rearrange the terms inside the brackets to write it in a more standard polynomial form:

dy/dx = (2x - 2)^4(1 - x3)3 [ -34x^3 + 24x^2 + 10]

Final Touches: Factoring Out Constants (Optional)

We can take our simplification one step further by factoring out a constant from the expression inside the brackets. Notice that all the coefficients (-34, 24, and 10) are divisible by 2. Let's factor out a 2:

dy/dx = 2(2x - 2)^4(1 - x3)3 [-17x^3 + 12x^2 + 5]

And there you have it! We've successfully differentiated the function and simplified the result.

The Final Answer

So, the derivative of y = (2x - 2)^5(1 - x3)4 is:

dy/dx = 2(2x - 2)^4(1 - x3)3 [-17x^3 + 12x^2 + 5]

Key Takeaways

This problem was a fantastic exercise in using the product rule and the chain rule together. Remember these key steps:

  1. Identify the product: Recognize when you need to use the product rule.
  2. Apply the chain rule: Know when to use the chain rule for composite functions.
  3. Simplify: Always simplify your answer as much as possible by factoring and combining like terms.

Differentiation problems can seem daunting, but breaking them down into smaller, manageable steps makes the process much easier. With practice, you'll become a pro at differentiating even the most complex functions! Keep up the great work, guys!