Derivative Of E^(6x)arcsin(x): A Step-by-Step Guide

by Andrew McMorgan 52 views

Hey math enthusiasts! Ever stumbled upon a function that looks like a mathematical beast, a combination of exponential and trigonometric functions, and wondered how to tame it? Today, we're going to break down one such beast: g(x) = e^(6x) * arcsin(x). Our mission? To find its derivative. Don't worry, guys, we'll make it super clear and easy to follow. Think of it as a fun puzzle rather than a daunting task. Ready to dive in?

Understanding the Challenge: The Product Rule

Before we jump into the solution, let's quickly recap the main tool we'll be using: the product rule. Remember, the product rule is our best friend when we need to differentiate a function that's the product of two other functions. In simpler terms, if we have a function h(x) that's defined as h(x) = u(x) * v(x), then its derivative h'(x) is given by:

h'(x) = u'(x) * v(x) + u(x) * v'(x)

In plain English, this means we take the derivative of the first function, multiply it by the second function, then add that to the first function multiplied by the derivative of the second function. Got it? Great! This is the key to unlocking our problem. Applying the product rule correctly is crucial for solving this type of derivative problem. It might seem a bit abstract now, but trust me, as we apply it to our specific function, it will become much clearer. Think of it as a recipe – you need the right ingredients (the product rule) to bake the perfect cake (the derivative).

Now, let's identify our u(x) and v(x) in the function g(x) = e^(6x) * arcsin(x). Clearly, we can see that e^(6x) and arcsin(x) are multiplied together. This directly implies that we can consider e^(6x) as our first function, u(x), and arcsin(x) as our second function, v(x). Breaking down the problem like this makes it less intimidating and more manageable. It's like dissecting a complex machine into its individual components to understand how each part works. By identifying u(x) and v(x), we've taken the first major step towards finding the derivative of g(x).

Breaking Down the Function: Identifying u(x) and v(x)

Okay, so looking at our function g(x) = e^(6x) * arcsin(x), we can see it's a product of two simpler functions. Let's break it down:

  • u(x) = e^(6x)
  • v(x) = arcsin(x)

See? Not so scary when we break it down. It's like taking a big, overwhelming task and dividing it into smaller, more manageable steps. This is a great strategy in mathematics and in life in general! Now that we've identified our u(x) and v(x), the next step is to find their individual derivatives. This is where our knowledge of basic differentiation rules comes in handy. Remember, the derivative of e^(x) is simply e^(x), but we have e^(6x) here, so we'll need to use the chain rule. And for arcsin(x), we should recall its standard derivative formula. Don't worry if you don't remember them off the top of your head; we'll go through them step by step.

Finding the derivatives of u(x) and v(x) is like gathering the necessary ingredients before we start cooking. Without these derivatives, we can't apply the product rule and find the derivative of g(x). So, let's roll up our sleeves and get those derivatives calculated! This is where the real fun begins, as we put our calculus skills to the test. Remember, each step we take brings us closer to the final solution, and the feeling of accomplishment when we solve a challenging problem is truly rewarding.

Finding the Derivatives of u(x) and v(x)

Now, let's find the derivatives of our functions. This is where things get a little more hands-on, but trust me, you've got this!

Finding u'(x)

We have u(x) = e^(6x). To find its derivative, u'(x), we need to use the chain rule. Remember the chain rule? It's essential for differentiating composite functions. The chain rule basically says that if you have a function within a function, you differentiate the outer function, keeping the inner function the same, and then multiply by the derivative of the inner function. In this case, our outer function is e^(x) and our inner function is 6x.

The chain rule formula is: d/dx [f(g(x))] = f'(g(x)) * g'(x)

Applying the chain rule here:

  • The derivative of e^(x) is e^(x), so the derivative of e^(6x) (keeping the inner function 6x the same) is e^(6x).
  • The derivative of the inner function, 6x, is simply 6.

Therefore, u'(x) = 6e^(6x)

See? The chain rule, once understood, is a powerful tool in our calculus arsenal. It allows us to tackle complex functions by breaking them down into simpler parts. This is a common theme in mathematics – taking something complex and making it manageable by using the right tools and techniques. Now that we've found u'(x), let's move on to v'(x). This will require us to recall the derivative of the arcsin function, which is a standard result in calculus.

Finding v'(x)

We have v(x) = arcsin(x). The derivative of arcsin(x) is a standard result that you might want to memorize (or have handy in your notes!). It's a formula that pops up quite often in calculus problems, especially those involving inverse trigonometric functions. If you don't remember it, no worries, we'll refresh our memory now.

The derivative of arcsin(x) is:

v'(x) = 1 / √(1 - x^2)

That's it! This is a direct application of a known derivative. Sometimes, calculus is about recognizing patterns and applying the right formulas. Now that we have both u'(x) and v'(x), we're halfway there! We have all the ingredients we need to finally apply the product rule. Think of this as the final preparation steps before we assemble everything. We've calculated the individual components, and now it's time to put them together to get our final answer. The anticipation is building!

Applying the Product Rule: Putting it All Together

Alright, we've done the groundwork. We've identified u(x) and v(x), and we've found their derivatives, u'(x) and v'(x). Now comes the moment of truth: applying the product rule! Remember the formula:

h'(x) = u'(x) * v(x) + u(x) * v'(x)

Let's plug in what we know:

  • u'(x) = 6e^(6x)
  • v(x) = arcsin(x)
  • u(x) = e^(6x)
  • v'(x) = 1 / √(1 - x^2)

So, substituting these into the product rule formula, we get:

g'(x) = (6e^(6x) * arcsin(x)) + (e^(6x) * (1 / √(1 - x^2)))

This is the derivative! We've done it! But, like a good chef, we can always try to make our dish even more presentable. In this case, we can simplify the expression a bit to make it look cleaner and easier to understand. Simplifying the derivative not only makes it look nicer but can also make it easier to work with in future calculations. So, let's take a moment to see if we can tidy things up.

Simplifying the Derivative: Making it Look Pretty

Our derivative looks like this:

g'(x) = (6e^(6x) * arcsin(x)) + (e^(6x) / √(1 - x^2))

We can simplify this by factoring out the common factor, e^(6x), from both terms:

g'(x) = e^(6x) * [6arcsin(x) + (1 / √(1 - x^2))]

And that's it! We've simplified the derivative as much as we can. This final form is much cleaner and easier to work with. It's like polishing a gemstone to reveal its true brilliance. By simplifying the expression, we've not only made it more aesthetically pleasing but also potentially more useful for any further analysis or calculations we might need to perform. This is a crucial step in many mathematical problems – simplifying the result to its most concise and understandable form.

Final Answer and Key Takeaways

So, the derivative of g(x) = e^(6x) * arcsin(x) is:

g'(x) = e^(6x) * [6arcsin(x) + (1 / √(1 - x^2))]

Woohoo! We conquered the beast! Give yourselves a pat on the back. This wasn't just about finding an answer; it was about understanding the process. We started with a complex function, broke it down into smaller parts, applied the product rule and chain rule, and finally simplified our result. This is the essence of problem-solving in mathematics – and in life!

Key Takeaways:

  • The Product Rule: This is your go-to tool when differentiating the product of two functions. Remember the formula: h'(x) = u'(x) * v(x) + u(x) * v'(x)
  • The Chain Rule: Essential for differentiating composite functions (a function within a function). Remember: d/dx [f(g(x))] = f'(g(x)) * g'(x)
  • Derivatives of Basic Functions: Knowing the derivatives of basic functions like e^(x) and arcsin(x) is crucial. Practice makes perfect!
  • Simplification: Always try to simplify your final answer. It makes it easier to work with and understand.

So, guys, the next time you encounter a function that seems intimidating, remember this journey. Break it down, use the right tools, and don't be afraid to simplify. You've got this! Keep practicing, keep exploring, and keep enjoying the beauty of mathematics! Until next time, happy differentiating!