Derivative Of P(x) = X² - X At X = 6
Hey guys! Today, we're diving into a simple yet essential concept in calculus: finding the derivative of a function at a specific point. We've got a quadratic function, p(x) = x² - x, and we need to figure out what its first derivative is when x equals 6. No sweat, right? Let's break it down step by step. Understanding derivatives is super useful because they tell us the instantaneous rate of change of a function. In simpler terms, it tells us how much the function is changing at any given point. This has tons of applications in physics, engineering, economics, and many other fields. So, getting a handle on derivatives is definitely worth your time. For those of you who are just starting out with calculus, don't worry! We'll take it slow and explain everything clearly. And for those of you who are already familiar with derivatives, this will be a good refresher. So, let's get started and find the derivative of p(x) = x² - x at x = 6!
Understanding the Problem
Before we jump into the math, let's make sure we understand what the problem is asking. We have a function, p(x) = x² - x. This is a simple quadratic function, which means it's a polynomial of degree 2. When we graph this function, we get a parabola. The derivative of a function, denoted as p'(x), gives us the slope of the tangent line to the graph of the function at any point x. So, when we find p'(6), we're finding the slope of the tangent line to the graph of p(x) at the point where x = 6. In other words, we're finding how much the function is changing at that specific point. The first derivative is a fundamental concept in calculus, representing the instantaneous rate of change of a function. It's like zooming in on the function's graph until you see a straight line – the slope of that line is the derivative at that point. So, to solve this problem, we first need to find the derivative of p(x), and then we need to plug in x = 6 to find the value of the derivative at that point. This will give us the slope of the tangent line at x = 6, which is what the problem is asking for. Remember, the derivative is not just a number; it's a function that tells us the rate of change of the original function at any point. That's why it's so powerful and useful in many different fields.
Finding the Derivative
Okay, now let's find the derivative of p(x) = x² - x. To do this, we'll use the power rule, which is a fundamental rule in calculus for finding derivatives of power functions. The power rule states that if f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. In other words, we multiply the coefficient by the exponent and then decrease the exponent by 1. Applying the power rule to x², we get 2x¹, which is just 2x. Applying the power rule to x, which is the same as x¹, we get 1x⁰, which is just 1. So, the derivative of x² - x is 2x - 1. Therefore, p'(x) = 2x - 1. This is the first derivative of the function p(x). Remember, the derivative is a function itself, and it tells us the slope of the tangent line to the graph of p(x) at any point x. Now that we have the derivative, we can find the value of the derivative at x = 6. In summary, finding the derivative involves applying rules like the power rule to each term in the function. It's a mechanical process, but it's important to understand what the derivative represents: the instantaneous rate of change of the function. This concept is crucial for understanding many other topics in calculus and its applications.
Evaluating the Derivative at x = 6
Alright, we've found that p'(x) = 2x - 1. Now, we need to evaluate this derivative at x = 6. This means we need to plug in x = 6 into the expression for p'(x). So, we have p'(6) = 2(6) - 1. This simplifies to p'(6) = 12 - 1, which gives us p'(6) = 11. Therefore, the value of the first derivative of the function p(x) = x² - x at x = 6 is 11. This means that the slope of the tangent line to the graph of p(x) at the point where x = 6 is 11. In other words, the function is increasing at a rate of 11 units for every 1 unit increase in x at that point. Evaluating the derivative at a specific point is a straightforward process, but it's important to understand what it represents: the instantaneous rate of change of the function at that point. This has many practical applications, such as finding the velocity of an object at a specific time or the marginal cost of producing a certain number of items. So, understanding how to evaluate derivatives is an essential skill in calculus.
Conclusion
So, there you have it! The first derivative of the function p(x) = x² - x at x = 6 is 11. We found this by first finding the derivative of the function, which is p'(x) = 2x - 1, and then evaluating this derivative at x = 6. This gave us p'(6) = 11. Remember, the derivative represents the instantaneous rate of change of the function, and evaluating it at a specific point tells us how much the function is changing at that point. Understanding derivatives is a fundamental concept in calculus, and it has many applications in various fields. By mastering this concept, you'll be well on your way to tackling more complex problems in calculus and its applications. Keep practicing, and you'll become a derivative pro in no time! Derivatives are like the speedometer of a function, telling us how fast it's changing at any given moment. They're used in everything from physics to economics to computer science, so learning how to work with them is a valuable skill. And remember, calculus isn't just about memorizing formulas; it's about understanding the underlying concepts and how they relate to the real world. So, keep exploring, keep questioning, and keep learning! Great job, guys! You've successfully navigated through finding the derivative of a simple function. Now you're one step closer to conquering calculus!