Derivative Of Arcsin(e^(3x)): Step-by-Step Solution

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Hey guys! Today, we're diving into a fun calculus problem: finding the derivative of y = arcsin(e^(3x)). This might look a bit intimidating at first, but don't worry! We'll break it down step by step so it's super easy to understand. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we're all on the same page. We're given the function y = arcsin(e^(3x)), and our mission, should we choose to accept it (and we do!), is to find dy/dx, which represents the derivative of y with respect to x. In simpler terms, we want to know how y changes as x changes. This involves using the chain rule, which is a fundamental concept in calculus for differentiating composite functions. Basically, a composite function is a function within a function, like what we have here – an exponential function inside an inverse trigonometric function. To master this, you should be familiar with the derivatives of basic functions, particularly the derivative of arcsin(u) and the derivative of e^(u). Understanding these building blocks will make the entire process smoother and more intuitive.

Why This Matters

You might be wondering, why bother with this? Well, derivatives are the bread and butter of calculus and have tons of applications in various fields. They help us understand rates of change, optimization problems, and much more. Plus, mastering these types of problems strengthens your calculus skills, which is always a win! Think of it like leveling up in a video game, but instead of defeating a boss, you're conquering a challenging math problem. Each step you take in solving this problem is like gaining experience points, making you a more formidable math warrior. And the more you practice, the better you'll get at recognizing patterns and applying the right techniques. So, let’s see what makes this problem tick and how to solve it.

Breaking Down the Solution

Okay, let's dive into the nitty-gritty. To find the derivative of y = arcsin(e^(3x)), we'll need to use the chain rule. The chain rule states that if we have a composite function y = f(g(x)), then the derivative dy/dx is given by dy/dx = f'(g(x)) * g'(x). In our case, we can think of arcsin as our outer function and e^(3x) as our inner function. This means we'll first differentiate the outer function (arcsin) with respect to its argument (e^(3x)), and then multiply that by the derivative of the inner function (e^(3x)) with respect to x. This might sound complex, but we’ll take it one step at a time to make sure it’s crystal clear.

Step 1: Identify the Outer and Inner Functions

As mentioned earlier, our outer function is arcsin(u) and our inner function is u = e^(3x). It's crucial to identify these correctly because this sets the stage for applying the chain rule effectively. Think of it as peeling an onion – you have layers, and we need to address each one in the right order. The outer layer, in this case, is the arcsin function, which encapsulates the inner exponential function. Recognizing this structure helps you understand how the derivatives will cascade down through the layers, ensuring you don’t miss any crucial steps.

Step 2: Differentiate the Outer Function

The derivative of arcsin(u) with respect to u is 1 / √(1 - u^2). This is a standard derivative that you should ideally have in your toolkit. If you don't remember it off the top of your head, it's a good idea to jot it down or add it to your cheat sheet. So, for our function, this becomes 1 / √(1 - (e(3x))2). Notice that we're just plugging in our inner function, e^(3x), wherever we see u in the derivative of arcsin(u). This is a key step in applying the chain rule – we’re taking the derivative of the outer function while keeping the inner function intact for now.

Step 3: Differentiate the Inner Function

Now, let's tackle the inner function, e^(3x). The derivative of e^(u) with respect to u is simply e^(u). However, we have e^(3x), so we need to apply the chain rule again! The derivative of 3x with respect to x is 3. Therefore, the derivative of e^(3x) with respect to x is 3e^(3x). This is another common derivative, so getting comfortable with exponential functions and their derivatives is essential for calculus success. Remember, the chain rule is like a set of Russian nesting dolls – each layer requires its own derivative, and we need to carefully peel each one to get to the core.

Step 4: Apply the Chain Rule

Time to put it all together! According to the chain rule, dy/dx = (derivative of outer function) * (derivative of inner function). We found that the derivative of the outer function is 1 / √(1 - (e(3x))2), and the derivative of the inner function is 3e^(3x). So, we multiply these together: dy/dx = (1 / √(1 - (e(3x))2)) * (3e^(3x)). This is where all our hard work pays off. We’ve taken the individual components and combined them in the correct order using the chain rule. It’s like assembling a puzzle – each piece (derivative) fits perfectly to create the final picture (the complete derivative).

Simplifying the Result

We're almost there! Now, let's simplify our result. We have dy/dx = (3e^(3x)) / √(1 - e^(6x)). This is the simplified form of the derivative. Sometimes, further simplification might be possible depending on the context of the problem, but this is generally considered the standard form. It’s always a good practice to simplify your results as much as possible, as it can make subsequent calculations easier and provide a clearer understanding of the function's behavior. In this case, we’ve combined the terms and expressed the derivative in a concise manner.

The Final Answer

So, after all that work, the final answer is:

dy/dx = (3e^(3x)) / √(1 - e^(6x))

And there you have it! We've successfully found the derivative of y = arcsin(e^(3x)). Give yourself a pat on the back – you’ve tackled a challenging calculus problem and come out victorious. This kind of problem is a staple in calculus courses, and mastering it will significantly boost your confidence and problem-solving skills. Remember, practice makes perfect, so don't hesitate to try similar problems to solidify your understanding. You've got this!

Key Takeaways

  • Chain Rule: The chain rule is your best friend when dealing with composite functions. Remember to differentiate the outer function and then multiply by the derivative of the inner function.
  • Derivatives of Standard Functions: Knowing the derivatives of basic functions like arcsin(u) and e^(u) is crucial. Keep those formulas handy!
  • Step-by-Step Approach: Break down the problem into smaller, manageable steps. This makes the entire process less daunting and helps prevent errors.
  • Simplify: Always try to simplify your result as much as possible. A simplified answer is easier to work with and interpret.

Practice Makes Perfect

To really nail this concept, try working through similar problems. For example, you could try finding the derivative of arcsin(e^(2x)) or arcsin(2e^(3x)). The more you practice, the more comfortable you'll become with the chain rule and the derivatives of various functions. Think of each problem as a new level in a game – each one you conquer makes you a stronger player. And just like in a game, don't be afraid to experiment with different strategies and approaches. Sometimes, the best way to learn is by trying things out and seeing what works.

Conclusion

Finding the derivative of y = arcsin(e^(3x)) might have seemed tricky at first, but by breaking it down step by step and using the chain rule, we were able to solve it successfully. Remember, calculus is all about practice and understanding the fundamental concepts. Keep practicing, and you'll become a calculus pro in no time! So, keep those pencils moving, and let's conquer more math problems together. You've got this!