Finding The Derivative: A Step-by-Step Guide

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Hey guys! Let's dive into the world of calculus and figure out the first derivative of the function f(x)=extsqrt(2x)−extcosxf(x) = ext{sqrt}(2x) - ext{cos} x. Don't worry, it's not as scary as it sounds. We'll break it down step-by-step to make sure everything is crystal clear. This is a super important concept in math, so understanding it will open up a lot of doors for you. Get ready to flex those brain muscles!

Understanding Derivatives

So, what exactly is a derivative? Think of it as a way to measure how a function changes. Specifically, it tells you the instantaneous rate of change of a function at a specific point. Imagine you're driving a car; the derivative would be like your speedometer, telling you how fast you're going at any given moment. In other words, it represents the slope of the tangent line to the function's graph at a particular point. That's a mouthful, right? Let's break it down further. Derivatives are fundamental to understanding the behavior of functions. They help us find things like maximum and minimum values, which are super useful in optimization problems. They're also essential in physics, engineering, and economics. Knowing how to calculate a derivative is like having a superpower! You can analyze and understand complex systems with ease.

Before we jump into the problem, let's quickly recap some basic rules that will come in handy. First, the power rule: If you have a function like xnx^n, its derivative is nxn−1nx^{n-1}. Easy peasy! Next, the derivative of $ ext{cos} x$ is −extsinx- ext{sin} x. Lastly, the derivative of a constant times a function is the constant times the derivative of the function. For example, the derivative of 2f(x)2f(x) is 2f′(x)2f'(x). Understanding these basics is the key to unlocking the problem. The derivative tells us the instantaneous rate of change of a function. It's not just a mathematical concept; it's a tool that allows us to model and understand the dynamic nature of the world around us. So, whether you are trying to understand the trajectory of a rocket, the growth of a population, or the flow of electricity, derivatives are your best friend! And remember, practice makes perfect. The more you work with derivatives, the more comfortable and confident you will become. Get ready to unleash your inner mathematician!

Breaking Down the Function

Alright, let's get back to our function: f(x)=extsqrt(2x)−extcosxf(x) = ext{sqrt}(2x) - ext{cos} x. To find its derivative, we'll take it one part at a time. This is where things get fun! First, we need to find the derivative of $ extsqrt}(2x)$. Think of $ ext{sqrt}(2x)$ as (2x)1/2(2x)^{1/2}. Now, using the power rule and the chain rule, which says the derivative of a composite function is the derivative of the outer function times the derivative of the inner function, here's how we do it The outer function is the power of 1/2, and the inner function is 2x2x. The derivative of $(2x)^{1/2$ is rac{1}{2}(2x)^{-1/2} imes 2. The '2' comes from the derivative of the inside function 2x2x. So, the derivative of $ ext{sqrt}(2x)$ simplifies to rac{1}{ ext{sqrt}(2x)}. Next up, we have −extcosx- ext{cos} x. As we mentioned earlier, the derivative of $ ext{cos} x$ is −extsinx- ext{sin} x. So, the derivative of −extcosx- ext{cos} x is −(−extsinx)=extsinx-(- ext{sin} x) = ext{sin} x. Remember to pay close attention to the signs – they can make a big difference! We are essentially applying the rules we've learned to each part of our function. The process involves identifying the different components, differentiating them individually, and then combining the results. This approach might seem complex at first, but with practice, it becomes second nature. Each step is a small victory, and by the end, you'll have successfully navigated the terrain of derivatives.

Now, let's take a closer look at the power rule. It's incredibly useful when dealing with expressions involving powers. To apply it, you first multiply the expression by the exponent and then subtract one from the exponent. The chain rule is another powerful tool, allowing us to deal with nested functions, like our $ ext{sqrt}(2x)$ example. By breaking down complex functions into simpler components, we can apply these rules methodically and arrive at the correct derivative. Keep in mind that understanding these rules is not just about memorization. It's about grasping the underlying principles. This kind of understanding will help you to tackle a wide variety of derivative problems with confidence and ease. So, as you continue to explore calculus, remember that each rule and concept builds upon the previous one, and with each solved problem, you deepen your understanding of the incredible power of mathematics.

Putting it All Together

Now we have the derivatives of both parts of our original function! We have rac{1}{ ext{sqrt}(2x)} from $ ext{sqrt}(2x)$ and $ ext{sin} x$ from −extcosx- ext{cos} x. So, to find the derivative of the whole function f(x)=extsqrt(2x)−extcosxf(x) = ext{sqrt}(2x) - ext{cos} x, we just combine these two results. So, the first derivative f′(x)f'(x) is rac{1}{ ext{sqrt}(2x)} + ext{sin} x. And there you have it, folks! We've successfully calculated the derivative. Give yourselves a pat on the back! It's like solving a puzzle, and it feels awesome when you get it right. Now, let's see which of the provided options matches our answer. The correct answer is (C) rac{1}{ ext{sqrt}(2x)} + ext{sin} x. How cool is that? You've not only solved the problem, but you've also learned some valuable math skills along the way. Feel the satisfaction of applying your new knowledge and confidently approaching similar problems in the future. Remember that the journey of learning math is a marathon, not a sprint, and with each problem solved, you are building a strong foundation for future mathematical endeavors.

Let's reiterate the key steps we took to solve this problem: We started by identifying the individual components of the function and then applied the power rule and the chain rule to each component. We then combined the results to find the final derivative. This methodical process is a cornerstone of calculus. It's a skill that you'll use throughout your mathematical journey. So, the next time you face a similar derivative problem, remember the steps we have discussed. Break down the problem into manageable parts, apply the appropriate rules, and take your time. With practice and persistence, you'll become a derivative master!

Conclusion

So there you have it, guys! We've successfully found the first derivative of f(x)=extsqrt(2x)−extcosxf(x) = ext{sqrt}(2x) - ext{cos} x. We learned about derivatives, the power rule, the chain rule, and how to apply them. Keep practicing, keep exploring, and keep the math fun! Remember, understanding calculus isn't just about memorizing formulas; it's about building a solid foundation of concepts that will help you solve problems in all sorts of areas. From understanding the rate of change to being able to graph and analyze functions, the ability to work with derivatives is invaluable. So, embrace the challenge, enjoy the journey, and never stop learning. You're doing great! Keep up the amazing work! If you have any questions or want to try some more examples, feel free to ask. Cheers!