Derivative Of H(x) = 6√x + 3∛x A Step-by-Step Guide

Hey guys! In this article, we're going to dive into the fascinating world of calculus and tackle a common problem: finding the derivative of a function. Specifically, we'll be working with the function h(x) = 6√x + 3∛x. Now, don't let the square roots and cube roots intimidate you! We'll break it down step by step, making sure everyone understands the process. This is a fundamental concept in calculus, and mastering it will open doors to more advanced topics. Whether you're a student just starting your calculus journey or someone looking to refresh your skills, this guide is for you. We'll cover everything from rewriting the function in a more manageable form to applying the power rule and simplifying the result. So, grab your pencils, and let's get started!

Before we jump into the calculations, let's take a closer look at the function h(x) = 6√x + 3∛x. The key here is understanding those radical expressions. Remember, a square root (√x) is the same as x raised to the power of 1/2, and a cube root (∛x) is the same as x raised to the power of 1/3. This is a crucial first step because it allows us to rewrite the function in a form that's easier to work with when finding derivatives. So, we can rewrite h(x) as h(x) = 6x^(1/2) + 3x^(1/3). See how much simpler that looks? Now we're dealing with exponents, which we have rules for! This transformation is super important because it sets the stage for applying the power rule, a cornerstone of derivative calculations. Recognizing these radical forms and converting them into exponential forms is a skill that will serve you well in many calculus problems. By doing this, we've turned what might have seemed like a daunting task into something much more approachable. Understanding this foundational step makes the rest of the process flow much more smoothly. So, remember, when you see roots, think exponents! It's your secret weapon for simplifying and solving these kinds of problems.

Okay, so we've established that h(x) = 6√x + 3∛x can be rewritten as h(x) = 6x^(1/2) + 3x^(1/3). This is a critical step because it transforms the function into a form where we can easily apply the power rule for differentiation. The power rule, as you might recall, states that if we have a term like x^n, its derivative is nx^(n-1). By converting the square root and cube root into fractional exponents, we've made the function perfectly suited for this rule. Think of it like preparing your ingredients before you start cooking – you need to have everything in the right form to create the final dish. In this case, rewriting the function is our ingredient prep, and the power rule is our cooking method. This step isn't just about making the function look different; it's about changing its form to unlock the tools we need to solve the problem. It's a fundamental technique in calculus, and you'll find yourself using it time and time again. So, make sure you're comfortable with this conversion process. It's the foundation upon which we'll build the rest of our solution. By mastering this, you're not just solving this specific problem; you're developing a key skill for tackling a wide range of calculus challenges.

Now for the fun part: applying the power rule! We've got our function in the perfect form: h(x) = 6x^(1/2) + 3x^(1/3). Remember the power rule? It says that the derivative of x^n is nx^(n-1). We're going to apply this rule to each term in our function. First, let's tackle 6x^(1/2). We multiply the coefficient (6) by the exponent (1/2), which gives us 3. Then, we subtract 1 from the exponent (1/2 - 1 = -1/2). So, the derivative of 6x^(1/2) is 3x^(-1/2). Next, let's do the same for 3x^(1/3). We multiply 3 by 1/3, which gives us 1. Then, we subtract 1 from the exponent (1/3 - 1 = -2/3). So, the derivative of 3x^(1/3) is x^(-2/3). Putting it all together, the derivative of h(x), which we'll call h'(x), is h'(x) = 3x^(-1/2) + x^(-2/3). See how the power rule makes this process so straightforward? It's a powerful tool that simplifies differentiation. This step is the heart of the problem, where we actually find the derivative. By carefully applying the power rule to each term, we've transformed the original function into its derivative. And that's a major accomplishment! So, take a moment to appreciate the elegance of the power rule and how it allows us to find the rate of change of our function.

We've found the derivative, but let's make it look even nicer! Our current derivative is h'(x) = 3x^(-1/2) + x^(-2/3). Now, mathematicians generally prefer to avoid negative exponents and fractional powers in their final answers. So, let's rewrite this using radicals and positive exponents. Remember that x^(-n) is the same as 1/x^n. So, 3x^(-1/2) can be rewritten as 3/(x^(1/2)), which is the same as 3/√x. Similarly, x^(-2/3) can be rewritten as 1/(x^(2/3)), which is the same as 1/(∛(x^2)). Therefore, we can rewrite h'(x) as h'(x) = 3/√x + 1/(∛(x^2)). This is a much cleaner and more standard way to express the derivative. Simplifying the derivative is important for a few reasons. First, it makes the answer easier to understand and interpret. Second, it often makes it easier to work with the derivative in further calculations. And third, it's simply good mathematical practice to present your answers in the simplest form possible. This final step is like putting the finishing touches on a masterpiece. We've done the hard work of finding the derivative, and now we're polishing it up to make it shine. By rewriting the derivative with positive exponents and radicals, we've made it more accessible and easier to use. So, always remember to simplify your answers – it's the hallmark of a skilled mathematician.

Alright, guys, we've done it! We've successfully found the derivative of h(x) = 6√x + 3∛x. After rewriting the function, applying the power rule, and simplifying the result, we've arrived at our final answer: h'(x) = 3/√x + 1/(∛(x^2)). This represents the instantaneous rate of change of the function h(x) at any given point x. It's a powerful result that tells us how the function is changing. So, give yourselves a pat on the back! You've navigated the world of radicals, exponents, and the power rule, and you've emerged victorious. This is a fundamental skill in calculus, and you've now added it to your toolkit. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a calculus pro in no time. This journey through finding the derivative highlights the beauty and power of calculus. We've taken a seemingly complex function and, through a series of logical steps, uncovered its rate of change. This ability to analyze how functions change is what makes calculus such a valuable tool in so many fields, from physics and engineering to economics and computer science. So, congratulations on mastering this concept, and keep exploring the amazing world of calculus!