Finding The Derivative Of G(x) = √(3x² + X⁵) A Calculus Guide

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Hey there, math enthusiasts! Today, we're diving into the exciting world of calculus to find the derivative of a fascinating function: G(x) = √(3x² + x⁵). Don't worry if this looks intimidating at first glance; we'll break it down step by step, making it super easy to understand. So, grab your pencils, and let's get started!

Understanding the Function and the Goal

Before we jump into the calculations, let's take a moment to understand what we're dealing with. Our function, G(x) = √(3x² + x⁵), is a composite function. This means it's a function within a function. The outer function is the square root, and the inner function is the polynomial 3x² + x⁵. Our goal is to find G'(x), which represents the derivative of G(x). The derivative tells us the instantaneous rate of change of the function at any given point. In simpler terms, it's the slope of the tangent line to the function's graph at that point. This concept is fundamental in calculus and has numerous applications in physics, engineering, economics, and many other fields. Finding the derivative of a composite function requires a special technique called the chain rule, which we'll explore in detail. The chain rule is a powerful tool that allows us to differentiate complex functions by breaking them down into simpler parts. It essentially states that the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function itself. This might sound a bit confusing now, but don't worry, we'll see how it works in practice. Understanding the chain rule is crucial for mastering calculus, as it's used extensively in various differentiation problems. So, let's keep this concept in mind as we move forward with our calculations. Remember, calculus is all about understanding change, and the derivative is our primary tool for quantifying that change. By finding the derivative of G(x), we'll gain valuable insights into its behavior and how it changes over time.

Applying the Chain Rule: A Step-by-Step Approach

Now comes the exciting part – applying the chain rule to find the derivative of G(x). As we discussed, the chain rule is our go-to technique for differentiating composite functions like this one. To make things crystal clear, let's break down the process into manageable steps:

1. Identify the Outer and Inner Functions: This is the crucial first step. As we mentioned earlier, our outer function is the square root, which we can rewrite as a power of 1/2: G(x) = (3x² + x⁵)^(1/2). The inner function is the polynomial inside the parentheses: 3x² + x⁵. Let's call the outer function u^(1/2) where u represents the inner function (3x² + x⁵). This substitution helps us visualize the chain rule more clearly. It's like peeling an onion – we're starting with the outermost layer and working our way inwards. Identifying the outer and inner functions correctly is essential for applying the chain rule effectively. A common mistake is to confuse the two, which can lead to incorrect results. So, always take a moment to clearly identify the outer and inner functions before proceeding. Remember, the outer function is the one that acts on the entire inner function. In our case, it's the square root acting on the polynomial. Once you've mastered this step, the rest of the process becomes much smoother.

2. Differentiate the Outer Function: Next, we need to find the derivative of the outer function with respect to u. Using the power rule (d/dx x^n = nx^(n-1)), the derivative of u^(1/2) is (1/2)u^(-1/2). This step involves applying a fundamental differentiation rule, the power rule. The power rule is one of the most basic and frequently used rules in calculus, so it's essential to have a solid understanding of it. It states that the derivative of x raised to the power of n is n times x raised to the power of n minus 1. In our case, we're applying the power rule to u^(1/2), where u is a function of x. This is where the chain rule comes into play. We're not just differentiating u^(1/2) with respect to u; we're differentiating it as part of a larger composite function. This means we need to multiply the result by the derivative of the inner function, as we'll see in the next step. But for now, let's focus on getting the derivative of the outer function correct. Remember, practice makes perfect. The more you apply the power rule, the more comfortable you'll become with it.

3. Differentiate the Inner Function: Now, let's differentiate the inner function, 3x² + x⁵, with respect to x. Again, we'll use the power rule. The derivative of 3x² is 6x, and the derivative of x⁵ is 5x⁴. So, the derivative of the inner function is 6x + 5x⁴. This step involves differentiating a polynomial, which is a straightforward application of the power rule. We simply apply the power rule to each term in the polynomial and add the results together. The derivative of 3x² is obtained by multiplying the coefficient (3) by the exponent (2) and reducing the exponent by 1, resulting in 6x^(2-1) = 6x. Similarly, the derivative of x⁵ is 5x^(5-1) = 5x⁴. The sum of these derivatives gives us the derivative of the entire inner function, which is 6x + 5x⁴. It's important to remember that the derivative of a sum is the sum of the derivatives. This property of differentiation makes it easy to differentiate polynomials term by term. Make sure you're comfortable with the power rule and the sum rule of differentiation before moving on. These rules are fundamental to calculus and will be used extensively throughout your studies.

4. Apply the Chain Rule Formula: The chain rule states that G'(x) = (derivative of outer function) * (derivative of inner function). Plugging in what we found, we get G'(x) = (1/2)u^(-1/2) * (6x + 5x⁴). Now, we substitute back u = 3x² + x⁵: G'(x) = (1/2)(3x² + x⁵)^(-1/2) * (6x + 5x⁴). This is the core of the chain rule in action. We're multiplying the derivative of the outer function, evaluated at the inner function, by the derivative of the inner function. This crucial step connects the rates of change of the outer and inner functions to give us the overall rate of change of the composite function. Make sure you understand why this multiplication is necessary. The derivative of the outer function tells us how the outer function changes with respect to its input (u), and the derivative of the inner function tells us how the inner function changes with respect to x. The product of these derivatives gives us how the outer function changes with respect to x, which is what we're ultimately trying to find. The chain rule is a powerful tool, but it's also easy to make mistakes if you're not careful. Always remember to multiply the derivatives of the outer and inner functions, and make sure you're evaluating the derivative of the outer function at the inner function.

Simplifying the Result

Okay, we've found the derivative, but it looks a bit messy. Let's simplify it to make it more presentable and easier to work with. Our current expression is G'(x) = (1/2)(3x² + x⁵)^(-1/2) * (6x + 5x⁴). Here's how we can clean it up:

1. Rewrite the Negative Exponent: A negative exponent means we have a reciprocal. So, (3x² + x⁵)^(-1/2) is the same as 1/√(3x² + x⁵). This step involves applying a basic rule of exponents: x^(-n) = 1/x^n. A negative exponent indicates that the base is in the denominator. In our case, the base is (3x² + x⁵), and the exponent is -1/2. So, we can rewrite (3x² + x⁵)^(-1/2) as 1/(3x² + x⁵)^(1/2). Remember that a fractional exponent represents a root. Specifically, an exponent of 1/2 represents a square root. So, (3x² + x⁵)^(1/2) is the same as √(3x² + x⁵). This rewriting allows us to get rid of the negative exponent and express the result in a more familiar form. It's often helpful to rewrite expressions with negative exponents to make them easier to understand and manipulate. This is a common technique used in calculus and other areas of mathematics. So, make sure you're comfortable with the rules of exponents and how to apply them.

2. Combine the Terms: Now we have G'(x) = (1/2) * (1/√(3x² + x⁵)) * (6x + 5x⁴). We can multiply the fractions together to get G'(x) = (6x + 5x⁴) / (2√(3x² + x⁵)). This step is a straightforward multiplication of fractions. We multiply the numerators together and the denominators together. The numerator is (1/2) * (6x + 5x⁴) = (6x + 5x⁴)/2, and the denominator is √(3x² + x⁵). Combining these, we get G'(x) = (6x + 5x⁴) / (2√(3x² + x⁵)). This simplified expression is much easier to work with than the previous one. It's often the case that simplifying an expression after differentiation is necessary to make it more useful. A simplified expression is easier to interpret, manipulate, and use in further calculations. So, always look for opportunities to simplify your results after applying differentiation rules.

3. Simplify Further (Optional): We can factor out a 2x from the numerator: G'(x) = 2x(3 + (5/2)x³) / (2√(3x² + x⁵)). Then, we can cancel the 2's: G'(x) = x(3 + (5/2)x³) / √(3x² + x⁵). This final simplification step involves factoring out a common factor from the numerator and canceling it with a factor in the denominator. In this case, we can factor out a 2x from the numerator, which gives us 2x(3 + (5/2)x³). Then, we can cancel the 2 with the 2 in the denominator, leaving us with G'(x) = x(3 + (5/2)x³) / √(3x² + x⁵). This is the most simplified form of the derivative. While this step is optional, it's often a good idea to simplify your results as much as possible. A simplified expression is not only easier to work with, but it also makes it easier to spot patterns and relationships. In some cases, a simplified expression may also be necessary for further calculations or applications. So, always consider whether you can simplify your results further after differentiation.

Final Answer

So, the derivative of G(x) = √(3x² + x⁵) is G'(x) = x(3 + (5/2)x³) / √(3x² + x⁵). Congratulations, you've successfully navigated the chain rule and found the derivative of a composite function! This is a significant achievement in your calculus journey. The chain rule is a fundamental concept in calculus, and mastering it will open doors to solving a wide range of differentiation problems. Remember, practice is key. The more you apply the chain rule to different functions, the more comfortable you'll become with it. Don't be afraid to tackle complex problems; break them down into smaller, manageable steps, just like we did in this article. And most importantly, have fun exploring the fascinating world of calculus!

Key Takeaways and Practice

Before we wrap up, let's highlight the key takeaways from this exercise:

  • The Chain Rule: This is your best friend for differentiating composite functions. Remember, it's all about the derivative of the outer function times the derivative of the inner function.
  • Step-by-Step Approach: Break down complex problems into smaller, manageable steps. This makes the process less daunting and reduces the chance of errors.
  • Simplification: Always simplify your results as much as possible. It makes them easier to understand and work with.

To solidify your understanding, try applying the chain rule to similar functions. For example, you could try finding the derivative of H(x) = (x³ + 2x)^(4/3) or J(x) = sin(x² + 1). Remember, practice makes perfect! Keep exploring, keep learning, and you'll become a calculus whiz in no time.

I hope this article has helped you understand how to find the derivative of G(x) = √(3x² + x⁵). If you have any questions, feel free to ask. Happy calculating!