Sign Of Derivative: (9-x^2)/x^2 Explained
Hey guys! Today, we're diving into a classic calculus problem: determining the sign of the derivative of the function f(x) = (9 - x^2) / x^2. This might sound intimidating, but trust me, we'll break it down step-by-step so it's super clear. Understanding the sign of a derivative is crucial because it tells us whether the original function is increasing or decreasing. So, let's get started and unravel this together!
Understanding the Basics of Derivatives
Before we jump into the specifics of our function, let's quickly recap what derivatives are all about. In simple terms, the derivative of a function at a particular point gives us the instantaneous rate of change of that function at that point. Think of it as the slope of the tangent line to the function's graph at that point. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing; and if it's zero, we've likely found a critical point (a potential maximum or minimum). Now, armed with this basic understanding, we can tackle our problem with confidence.
When we talk about derivatives, it's super important to remember that we're looking at how a function changes. The derivative, often written as f'(x) or dy/dx, is like a magnifying glass for the function's behavior at a tiny, tiny scale. A positive derivative means the function is climbing uphill – as x increases, y also increases. A negative derivative means the function is sliding downhill – as x increases, y decreases. And when the derivative is zero, it's like the function is taking a brief pause at the top of a hill or the bottom of a valley. These pauses are critical points, and they're super helpful for understanding the overall shape of a function's graph. Derivatives help us pinpoint exactly where these changes happen, giving us a super detailed understanding of the function's ups and downs. So, with this knowledge in our toolkit, let's get to figuring out the sign of the derivative for our particular function – it's gonna be a fun ride!
Finding the Derivative of f(x) = (9 - x^2) / x^2
Okay, the first step in figuring out the sign of the derivative is, well, finding the derivative itself! We have f(x) = (9 - x^2) / x^2. To make things easier, we can rewrite this function as f(x) = 9/x^2 - 1. This simplifies our job because we can now use the power rule for differentiation. Remember, the power rule states that if we have a term like x^n, its derivative is n*x^(n-1). So, let's apply this to our function.
First, rewrite 9/x^2 as 9x^(-2). Now, applying the power rule, the derivative of 9x^(-2) is -18x^(-3), which can be written as -18/x^3. The derivative of the constant term -1 is simply 0. Therefore, the derivative of our function, f'(x), is -18/x^3. We've successfully found the derivative! Now, the next step is to figure out where this derivative is positive, negative, or zero. This will give us insights into the behavior of the original function.
Finding the derivative might sound like a complex task, but once you break it down into simpler steps, it becomes much more manageable. In our case, rewriting the original function made a huge difference. It allowed us to use the power rule, which is a straightforward way to differentiate terms of the form x^n. The power rule is a cornerstone of calculus, and mastering it is key to tackling a wide range of differentiation problems. When we applied it to 9x^(-2), we multiplied the coefficient (9) by the exponent (-2) and then decreased the exponent by 1, resulting in -18x^(-3). This kind of step-by-step approach is what makes calculus less daunting. Now that we've got our derivative, f'(x) = -18/x^3, we're in a great position to analyze its sign and understand the original function's behavior. So, let's move on to the next step and see what we can uncover!
Analyzing the Sign of the Derivative
Now that we have the derivative, f'(x) = -18/x^3, we need to figure out its sign for different values of x. Remember, the sign of the derivative tells us whether the original function is increasing or decreasing. The key here is to look at the factors that make up the derivative.
We have a constant -18 in the numerator, which is always negative. The denominator is x^3, which can be positive or negative depending on the value of x. If x is positive, then x^3 is positive. If x is negative, then x^3 is negative. So, the sign of f'(x) depends entirely on the sign of x^3. When x > 0, x^3 is positive, and f'(x) = -18/x^3 is negative. This means the original function f(x) is decreasing when x is positive. When x < 0, x^3 is negative, and f'(x) = -18/x^3 is positive. This means the original function f(x) is increasing when x is negative. At x = 0, the derivative is undefined because we'd be dividing by zero.
Analyzing the sign of the derivative is like being a detective, piecing together clues to understand a function's behavior. The fact that our derivative, f'(x) = -18/x^3, has a simple form is a huge advantage. We can clearly see that the numerator is always negative, so the sign of the entire derivative hinges on the sign of the denominator, x^3. This is where our understanding of basic algebra comes into play. We know that cubing a positive number results in a positive number, and cubing a negative number gives us a negative number. This direct relationship between x and x^3 is the key to unlocking the mystery of f'(x)'s sign. By considering these two cases – x > 0 and x < 0 – we can definitively say where the original function is increasing and where it's decreasing. Remember, this kind of analysis is not just about crunching numbers; it's about developing a deep, intuitive understanding of how functions behave. So, with our analysis complete, we're one step closer to fully grasping the characteristics of f(x) = (9 - x^2) / x^2.
Intervals of Increase and Decrease
From our analysis, we know that f'(x) is positive when x < 0 and negative when x > 0. This translates directly to the intervals where the original function f(x) is increasing and decreasing. The function f(x) is increasing on the interval (-∞, 0) and decreasing on the interval (0, ∞). It's important to note that x = 0 is not included in these intervals because the function and its derivative are undefined at x = 0.
Understanding intervals of increase and decrease is a fundamental aspect of calculus. It's like having a roadmap for a function's journey. We've discovered that our function, f(x) = (9 - x^2) / x^2, is on an uphill climb as we move from negative infinity towards zero, and then it switches gears and starts heading downhill as we move from zero towards positive infinity. The fact that the function is undefined at x = 0 is a critical detail. It tells us that there's a break in the function's graph at this point, and we can't simply connect the increasing and decreasing parts. This kind of break, or discontinuity, is an important feature to recognize when analyzing functions. So, armed with this knowledge of where f(x) is increasing and decreasing, we're building a comprehensive picture of its behavior. We're not just finding answers; we're developing a visual and intuitive sense of how the function changes, which is what calculus is all about!
Conclusion
So, to recap, we found that the derivative of f(x) = (9 - x^2) / x^2 is f'(x) = -18/x^3. By analyzing the sign of the derivative, we determined that f(x) is increasing on the interval (-∞, 0) and decreasing on the interval (0, ∞). Remember, guys, understanding the sign of the derivative is a powerful tool in calculus. It allows us to understand the behavior of a function, find critical points, and sketch its graph. Keep practicing, and you'll master these concepts in no time!
This journey through the derivative of f(x) = (9 - x^2) / x^2 has been a fantastic example of how calculus helps us unravel the mysteries of functions. We started with a function that might have seemed a bit complex, but by systematically applying the rules of differentiation and carefully analyzing the sign of the derivative, we've gained a deep understanding of its behavior. This process – from finding the derivative to interpreting its sign – is a cornerstone of calculus. It's not just about getting the right answer; it's about developing a way of thinking that allows you to explore and understand the intricate world of mathematical functions. So, keep those pencils sharp, keep asking questions, and keep exploring the fascinating world of calculus – there's always more to discover! And remember, every problem you solve is a step forward in your mathematical journey. Let's keep the momentum going and tackle the next challenge with enthusiasm and curiosity!