Derivative Of X(arctan(x))^2: A Step-by-Step Guide

Hey guys! Today, we're diving into a fun calculus problem: finding the derivative of the function x(arctan(x))^2. This might seem a bit intimidating at first, but don't worry, we'll break it down step by step. We'll be using a couple of key calculus rules, namely the product rule and the chain rule, so make sure you're comfy with those. Let's get started and make calculus a little less scary, one derivative at a time!

Understanding the Function

Before we jump into the differentiation process, let's make sure we're all on the same page about what the function x(arctan(x))^2 actually represents. At its core, this is a product of two simpler functions: the linear function x and the square of the arctangent function, (arctan(x))^2. The arctangent function, often written as arctan(x) or tan⁻¹(x), is the inverse of the tangent function. It essentially answers the question, "What angle has a tangent of x?" So, when we see arctan(x), we're thinking about an angle whose tangent is x. Now, we're not just dealing with arctan(x) on its own; we're squaring it and then multiplying the result by x. This combination of functions is what makes finding the derivative a bit more involved, but also more interesting! To tackle this, we need to remember our differentiation rules, especially the product rule, which is crucial when differentiating a product of two functions, and the chain rule, which comes into play when dealing with composite functions like (arctan(x))^2. Recognizing these components is the first step in conquering this calculus challenge. So, with our function clearly in mind, we're ready to move on to the next step: applying the rules of differentiation to unravel its derivative.

Applying the Product Rule

Alright, let's get our hands dirty with some actual calculus! As we discussed, our function, x(arctan(x))^2, is a product of two functions: x and (arctan(x))^2. This is a clear signal that we need to employ the product rule. For those who need a quick refresher, the product rule states that the derivative of two functions, say u(x) and v(x), multiplied together is given by: (u(x)v(x))' = u'(x)v(x) + u(x)v'(x). In simpler terms, it's the derivative of the first function times the second function, plus the first function times the derivative of the second function. So, in our case, we can consider u(x) = x and v(x) = (arctan(x))^2. Now, we need to find the derivatives of u(x) and v(x) separately. The derivative of u(x) = x is straightforward: u'(x) = 1. For v(x) = (arctan(x))^2, we'll need to use another important rule, the chain rule, which we'll tackle in the next section. But for now, let's just acknowledge that we'll need to find v'(x) and then plug everything into the product rule formula. Applying the product rule is a fundamental step in differentiating complex functions like this one. It allows us to break down the problem into smaller, more manageable parts. Once we've found the derivatives of our individual functions, we can piece them together to find the overall derivative. So, let's keep this momentum going and dive into the chain rule to find the derivative of (arctan(x))^2.

Using the Chain Rule

Now, let's focus on finding the derivative of (arctan(x))^2, which we identified as v(x) in the previous section. This is where the chain rule comes to our rescue. The chain rule is essential when we're dealing with composite functions – that is, functions within functions. In our case, we have the arctangent function being squared, which is a classic example of a composite function. The chain rule states that if we have a function y = f(g(x)), then its derivative is given by: dy/dx = f'(g(x)) * g'(x). In simpler terms, we take the derivative of the outer function, keeping the inner function as is, and then multiply by the derivative of the inner function. So, for v(x) = (arctan(x))^2, we can think of the outer function as f(u) = u^2 and the inner function as g(x) = arctan(x). First, let's find the derivative of the outer function with respect to u: f'(u) = 2u. Now, we substitute g(x) back in for u, giving us 2arctan(x). Next, we need to find the derivative of the inner function, g(x) = arctan(x). The derivative of arctan(x) is a standard result that you might want to memorize: g'(x) = 1 / (1 + x^2). Finally, we apply the chain rule formula: v'(x) = f'(g(x)) * g'(x) = 2arctan(x) * (1 / (1 + x^2)). So, we've successfully found the derivative of (arctan(x))^2 using the chain rule! This is a significant step in solving our overall problem. Now that we have the derivatives of both u(x) and v(x), we're ready to plug them back into the product rule formula and find the final derivative.

Combining the Results

Okay, we've done the groundwork, and now it's time for the grand finale: combining everything we've found to get the derivative of x(arctan(x))^2. Remember, we used the product rule to break down the problem, and we needed the chain rule to find the derivative of (arctan(x))^2. Let's recap what we've got: We identified u(x) = x and v(x) = (arctan(x))^2. We found that u'(x) = 1. And, using the chain rule, we determined that v'(x) = 2arctan(x) * (1 / (1 + x^2)). Now, let's plug these pieces into the product rule formula: (u(x)v(x))' = u'(x)v(x) + u(x)v'(x). Substituting our results, we get: (x(arctan(x))^2)' = 1 * (arctan(x))^2 + x * [2arctan(x) * (1 / (1 + x^2))]. Let's simplify this a bit: (x(arctan(x))^2)' = (arctan(x))^2 + (2xarctan(x)) / (1 + x^2). And there you have it! We've successfully found the derivative of x(arctan(x))^2. It might look a bit complex, but we arrived here by systematically applying the product and chain rules. This is a great example of how breaking down a problem into smaller steps can make even the trickiest calculus challenges manageable. Now that we have the derivative, we can use it for various applications, such as finding critical points, analyzing the function's behavior, or even solving optimization problems. But for now, let's bask in the glory of having conquered this derivative!

Final Derivative and Simplification

So, we've arrived at the derivative of our function, x(arctan(x))^2, which we found to be (arctan(x))^2 + (2xarctan(x)) / (1 + x^2). This is a perfectly valid answer, but in mathematics, we often like to simplify our results as much as possible to make them cleaner and easier to work with. While this derivative might not have a dramatically simpler form, we can still do a little algebraic maneuvering to make it look a bit more elegant. One common approach is to combine the terms into a single fraction. To do this, we need a common denominator, which in this case is (1 + x^2). So, let's rewrite our derivative with this common denominator: [(arctan(x))^2 * (1 + x^2) + 2xarctan(x)] / (1 + x^2). Now, we can distribute the (arctan(x))^2 term in the numerator: [(arctan(x))^2 + x^2(arctan(x))2 + 2xarctan(x)] / (1 + x^2). This is about as simplified as we can reasonably get without further algebraic manipulation that might not necessarily improve the expression significantly. The final simplified form, [(arctan(x))^2 + x^2(arctan(x))2 + 2xarctan(x)] / (1 + x^2), is a concise representation of the derivative. It showcases the key components and their relationships, making it easier to analyze and use in further calculations. Simplifying derivatives is not just about aesthetics; it's about making the result more accessible and useful for whatever comes next. Whether it's finding critical points, determining concavity, or solving related rates problems, a simplified derivative is your best friend. So, with our simplified derivative in hand, we've truly completed our mission of finding the derivative of x(arctan(x))^2. Give yourself a pat on the back – you've earned it!

Conclusion

Alright, guys, we've reached the end of our journey to find the derivative of x(arctan(x))^2. We've navigated the twists and turns of the product rule and the chain rule, and we emerged victorious with a simplified derivative in hand. This problem is a fantastic example of how calculus can seem daunting at first, but by breaking it down into smaller, manageable steps, we can conquer even the most complex functions. We started by understanding the function itself, recognizing it as a product of x and (arctan(x))^2. This led us to the product rule, which told us how to differentiate a product of two functions. Then, we zoomed in on (arctan(x))^2 and realized we needed the chain rule to handle this composite function. We carefully applied the chain rule, found the derivative of arctan(x), and plugged everything back into our product rule formula. Finally, we simplified our result to arrive at a clean and usable derivative. The key takeaway here isn't just the answer itself, but the process we followed. Calculus is all about understanding the rules and knowing when and how to apply them. Practice makes perfect, so the more you work through problems like this, the more confident you'll become. Remember, every derivative you find is a step forward in your calculus journey. So, keep exploring, keep learning, and never be afraid to tackle a challenging problem. You've got this! And who knows, maybe our next adventure will be even more exciting!