Hey guys! Let's dive into the fascinating world of calculus and figure out how to differentiate sin³x cos³x with respect to x. This might seem a little intimidating at first, but trust me, we'll break it down step by step to make it super easy to understand. We'll be using some fundamental differentiation rules, including the chain rule and the product rule. By the end of this guide, you'll be able to confidently tackle this type of problem. So, grab your pencils and let's get started!

Understanding the Problem and Prerequisites

First things first, what exactly are we dealing with? We have the function f(x) = sin³x cos³x. Our goal is to find its derivative, denoted as f'(x) or dy/dx. This represents the instantaneous rate of change of the function with respect to x. Before we jump in, let's brush up on a few key concepts. You should be familiar with the following:

• Basic Differentiation Rules: The power rule (d/dx xⁿ = nx^(n-1)), the derivative of sin(x) (cos(x)), and the derivative of cos(x) (-sin(x)).
• Product Rule: If we have a function in the form of u(x)v(x), the derivative is given by (uv)' = u'v + uv'.
• Chain Rule: When differentiating a composite function, like f(g(x)), the derivative is f'(g(x)) * g'(x).

If you're a bit rusty on these, don't worry! We'll review them briefly as we go along. Think of it like this: differentiation is all about finding how a function changes. The derivative gives us this information. The chain and product rules are our main tools here, and we'll apply them strategically to solve the problem.

Now, let's break down the problem further. We're essentially working with a product of two functions: sin³x and cos³x. Each of these also involves a power and a trigonometric function, which means the chain rule will be our friend. So, buckle up, because we're about to apply these rules to solve the main problem.

Breaking Down the Function: sin³x and cos³x

To make things easier, let's rewrite our original function: f(x) = sin³x cos³x. This is the same as (sinx)³ * (cosx)³. Now, consider each part separately. We have (sin x)³ and (cos x)³. These are composite functions, so we'll need to use the chain rule to find their derivatives.

Let's start with (sin x)³. Applying the chain rule, we can rewrite it as (u)³ where u = sin x. So, the derivative of u³ is 3u². Now we need to multiply by the derivative of u which is cos x. Hence, the derivative of (sin x)³ = 3sin²x cos x.

Similarly, for (cos x)³, using the chain rule again with u = cos x, we find its derivative. The derivative of u³ = 3u², so we get 3cos²x. Then, we need to multiply by the derivative of u which is -sin x. Thus, the derivative of (cos x)³ = -3cos²x sin x.

So, we now have the derivatives of sin³x and cos³x individually. These components are essential for the next step, which will involve applying the product rule to the original function. We're moving closer to finding the complete derivative of the entire expression, keeping in mind the structure of the problem that calls for a combined approach using both the chain rule and the product rule.

Applying the Product Rule: Putting It All Together

Alright, now that we've found the derivatives of sin³x and cos³x individually, it's time to bring everything together using the product rule. Recall the product rule: If f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x).

In our case, let u(x) = sin³x and v(x) = cos³x. We already know:

• u'(x) = 3sin²x cos x
• v'(x) = -3cos²x sin x

Now, let's plug these into the product rule formula:

f'(x) = (3sin²x cos x)(cos³x) + (sin³x)(-3cos²x sin x)

Simplify this expression:

f'(x) = 3sin²x cos⁴x - 3sin⁴x cos²x

And there you have it, folks! The derivative of sin³x cos³x. We've combined the chain rule (used to differentiate sin³x and cos³x) and the product rule (used to differentiate the product of these functions) to solve this problem. The result is simplified to its final form and it is the complete answer for differentiating the original expression.

Simplifying the Derivative: Further Steps

We have successfully found the derivative, but we can simplify it even further. Let's factor out the common terms from our derivative: f'(x) = 3sin²x cos⁴x - 3sin⁴x cos²x. Both terms have 3, sin²x, and cos²x in common. Factoring these out, we get:

f'(x) = 3sin²x cos²x (cos²x - sin²x)

This is a cleaner, more concise form of the derivative. Remember the trigonometric identity cos²x - sin²x = cos 2x. We can use this to simplify it further:

f'(x) = 3sin²x cos²x cos 2x

We could even go one step further using the double-angle identity: sin 2x = 2sinx cosx. Squaring both sides, we get sin² 2x = 4sin²x cos²x. Therefore, sin²x cos²x = (1/4)sin² 2x. Substituting this into our derivative, we get:

f'(x) = (3/4)sin² 2x cos 2x

Although the above is a valid form, the derivative: f'(x) = 3sin²x cos²x (cos²x - sin²x) is also a completely valid answer. It demonstrates how to manipulate and simplify trigonometric functions to a cleaner, more useful result. This simplifies the expression and makes it easier to analyze and use in further mathematical operations, like finding critical points or analyzing the behavior of the original function.

Conclusion: Recap and Key Takeaways

Well, that's a wrap! We've successfully differentiated sin³x cos³x with respect to x. Here's a quick recap of what we did:

1. Understood the Problem: We identified the function and the goal (finding the derivative).
2. Reviewed Prerequisites: We refreshed our knowledge of basic differentiation rules, the product rule, and the chain rule.
3. Broke Down the Function: We identified the composite functions (sin³x and cos³x).
4. Applied the Chain Rule: We found the derivatives of sin³x and cos³x individually.
5. Applied the Product Rule: We combined the individual derivatives using the product rule.
6. Simplified the Result: We simplified the derivative using trigonometric identities.

The key takeaways here are:

• The Chain Rule is crucial for differentiating composite functions.
• The Product Rule is essential for differentiating the product of two functions.
• Trigonometric Identities can be used to simplify and rewrite derivatives.

Remember, practice makes perfect! Try working through similar problems on your own to solidify your understanding. You can start by differentiating slightly different functions or modifying the original one. Keep practicing, and you'll become a pro at differentiation in no time. If you have any more questions, feel free to ask. Cheers!