Finding The Derivative Of Y = 3 Over Square Root Of X Plus 4

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Hey guys! Today, we're diving into a super common calculus problem: finding the derivative. Specifically, we're going to tackle the function y = 3/√(x+4). This might seem a bit intimidating at first, but don't worry, we'll break it down step by step. Derivatives are a fundamental concept in calculus, representing the instantaneous rate of change of a function. Understanding how to find derivatives is crucial for many applications, including optimization problems, physics, and engineering. This particular problem involves the application of the power rule and the chain rule, which are essential tools in differential calculus. We will explore these rules in detail as we solve the problem, ensuring a clear and comprehensive understanding of the process. So, let’s grab our metaphorical calculus tools and get started! We'll go through each step meticulously, making sure you understand the why behind the how. By the end of this article, you'll not only know how to solve this specific problem but also have a solid foundation for tackling similar derivative challenges. Remember, practice makes perfect, so don't hesitate to try out more examples on your own after we're done here. Let's get started and conquer this derivative together!

Understanding the Problem

Before we jump into the solution, let's make sure we fully understand the question. We need to find dy/dx, which means we're looking for the derivative of y with respect to x. In simpler terms, we want to know how y changes as x changes. Our function is y = 3/√(x+4). This function involves a fraction and a square root, which means we'll likely need to use a combination of differentiation rules. Specifically, we’ll be using the power rule and the chain rule. The power rule helps us differentiate terms raised to a power, while the chain rule is essential for differentiating composite functions (functions within functions). In our case, we have a function (x+4) inside a square root, and the entire square root expression is in the denominator of a fraction. This nested structure is a classic indicator that the chain rule will be necessary. Visualizing the problem is also helpful. Imagine the graph of the function y = 3/√(x+4). The derivative at any point on this graph represents the slope of the tangent line at that point. Finding the derivative allows us to analyze how the slope changes across the graph, which can reveal important information about the function's behavior, such as where it's increasing, decreasing, or reaching a maximum or minimum value. Understanding the problem thoroughly is the first step towards finding the correct solution. So, now that we've dissected the question and identified the tools we'll need, let's move on to the solution itself.

Step 1: Rewrite the Function

The first trick to make this derivative easier to handle is to rewrite the function. Instead of dealing with a square root in the denominator, let's express it using exponents. Remember that the square root of something is the same as raising it to the power of 1/2. Also, a term in the denominator can be moved to the numerator by changing the sign of its exponent. So, we can rewrite y = 3/√(x+4) as y = 3(x+4)^(-1/2). This transformation is crucial because it sets us up perfectly for using the power rule and the chain rule. By expressing the function in this form, we avoid the complexities of the quotient rule, which would be necessary if we left the function as a fraction. Rewriting the function might seem like a small step, but it significantly simplifies the differentiation process. It's a common technique in calculus to manipulate functions into a more manageable form before applying differentiation rules. This step demonstrates the importance of algebraic manipulation in calculus, highlighting how a simple rewrite can transform a seemingly complex problem into a straightforward one. Now that we have our function in a more user-friendly format, we are ready to move on to the next step: applying the power rule and the chain rule to find the derivative. This is where the real calculus magic happens, so let's dive in!

Step 2: Apply the Power Rule and Chain Rule

Alright, now for the fun part: differentiating! We'll use the power rule and the chain rule. The power rule states that if you have a term x^n, its derivative is n * x^(n-1). The chain rule, on the other hand, is used when you have a function inside another function. It says that the derivative of f(g(x)) is f'(g(x)) * g'(x). In our case, we have y = 3(x+4)^(-1/2). Let's break it down:

  1. Bring down the exponent: Multiply the entire term by the exponent (-1/2): 3 * (-1/2) = -3/2
  2. Keep the inside function: The (x+4) part stays the same for now.
  3. Subtract 1 from the exponent: (-1/2) - 1 = -3/2. So, we now have (x+4)^(-3/2).
  4. Multiply by the derivative of the inside function: The derivative of (x+4) is simply 1.

Putting it all together, we get dy/dx = (-3/2)(x+4)^(-3/2) * 1. This might look a bit messy, but we're almost there! Remember, the chain rule is crucial here because we have a function (x+4) nested inside another function (the power of -1/2). The power rule helps us deal with the exponent, while the chain rule ensures we account for the derivative of the inner function. This step is the heart of the problem, where we actually apply the differentiation rules to find the derivative. It's a great example of how calculus rules work together to solve complex problems. Now, let's move on to the final step and simplify our result to get a cleaner and more presentable answer.

Step 3: Simplify the Result

Our current derivative looks like this: dy/dx = (-3/2)(x+4)^(-3/2). To simplify, let's get rid of the negative exponent and rewrite it in a more familiar form. A negative exponent means we can move the term to the denominator and make the exponent positive. So, (x+4)^(-3/2) becomes 1/(x+4)^(3/2). Now we have dy/dx = (-3/2) * (1/(x+4)^(3/2)) which simplifies to dy/dx = -3 / (2(x+4)^(3/2)). We can also rewrite the fractional exponent as a square root and a cube. Remember that (x+4)^(3/2) is the same as √((x+4)^3) or (√(x+4))^3. So, our final simplified derivative is dy/dx = -3 / (2√((x+4)^3)). This is the derivative of our original function, expressed in a clean and understandable way. Simplifying the result is an important step in calculus. It not only makes the answer more presentable but also makes it easier to work with in further calculations or applications. By removing negative exponents and rewriting fractional exponents as radicals, we transform the derivative into a form that is both mathematically correct and aesthetically pleasing. This final step completes our journey of finding the derivative of y = 3/√(x+4). We started by understanding the problem, then rewrote the function, applied the power rule and chain rule, and finally simplified the result. Congratulations, you've successfully navigated a classic calculus problem!

Final Answer

So, after all that work, we've arrived at our final answer: dy/dx = -3 / (2√((x+4)^3)). This is the derivative of the function y = 3/√(x+4). We've successfully found how the function changes with respect to x. Remember, this derivative represents the slope of the tangent line at any point on the graph of the original function. It's a powerful tool for analyzing the behavior of the function, such as finding its increasing and decreasing intervals, as well as its maximum and minimum values. Understanding the meaning of the derivative is just as important as knowing how to calculate it. The derivative provides valuable insights into the function's characteristics and its applications in various fields. So, not only have we found the answer, but we've also reinforced our understanding of what the derivative actually represents. This is the essence of calculus – connecting the mathematical operations with their real-world interpretations. With this problem under our belts, we've strengthened our calculus skills and are better equipped to tackle more challenging problems in the future. Keep practicing, and you'll become a derivative-finding pro in no time!

Conclusion

Awesome job, guys! We've walked through finding the derivative of y = 3/√(x+4) step by step. We started by rewriting the function to make it easier to differentiate, then applied the power rule and the chain rule, and finally simplified our result. The final answer, dy/dx = -3 / (2√((x+4)^3)), represents the instantaneous rate of change of the function. This process highlights the importance of understanding the underlying rules and techniques of calculus, as well as the power of algebraic manipulation in simplifying complex problems. Remember, calculus is like learning a new language – it takes practice and patience to become fluent. Don't be discouraged if you find it challenging at first. The more problems you solve, the more comfortable you'll become with the concepts and techniques. This problem is a great example of how calculus can be used to analyze the behavior of functions. By finding the derivative, we gain valuable information about the function's slope, which helps us understand its increasing and decreasing intervals, as well as its maximum and minimum values. So, keep exploring, keep practicing, and keep pushing your calculus skills to the next level. You've got this! And remember, every derivative you conquer brings you one step closer to mastering calculus.