Hey guys! Let's dive into a classic calculus problem: finding dy/dx when we're given x and y in terms of another variable, in this case, t. This is a super common scenario, so understanding how to tackle it is a total game-changer. We'll break it down into easy-to-follow steps, making sure you grasp the concepts, even if you're just starting out. We're going to solve the problem where x = at² and y = 2at. Let's get started!

Understanding the Problem: Parametric Equations

Alright, so what does it even mean when we say x = at² and y = 2at? These are called parametric equations. Basically, instead of y being directly defined in terms of x (like in a regular equation), both x and y are defined in terms of a third variable, t in this case, which is called a parameter. Think of t as a time variable, and x and y are positions that change as time goes on. Our goal is to find dy/dx, which represents the rate of change of y with respect to x. In other words, how does y change as x changes? This gives us the slope of the curve defined by these parametric equations. The key here is to realize that we can't directly find dy/dx without using the chain rule. Instead, we have to find dy/dt and dx/dt first, and then combine them.

This is where things get interesting, and this is why knowing the chain rule is crucial. The chain rule is the tool that lets us connect the rates of change. It's like saying, "If I know how y changes with respect to t, and I know how t changes with respect to x, then I can figure out how y changes with respect to x." This is the foundation upon which we will solve our problem. The beauty of the chain rule is that it allows us to break down complex derivatives into smaller, more manageable parts. By understanding these concepts, you're not just solving a problem, you're building a deeper understanding of how calculus works. So, let's roll up our sleeves and get our hands dirty with some derivatives!

To really get this, imagine t as the control knob. As you turn the knob (change t), both x and y change, and dy/dx tells us how y moves compared to x. This parametric representation is super useful in all sorts of applications, from physics to computer graphics, and understanding how to deal with it is a solid step toward calculus mastery. We can think of it like this: dy/dx is the slope, dy/dt tells us how fast y is changing, and dx/dt tells us how fast x is changing. Putting it all together, we're really examining the relationship between the changes of these variables with respect to t.

Step-by-Step Solution: Finding dy/dx

Alright, let's get down to business and find that dy/dx. We'll break this down into digestible steps. First, we need to find dy/dt and dx/dt. Then, we'll use a simple formula to put it all together. Here we go!

Step 1: Find dx/dt

We are given x = at². To find dx/dt, we need to differentiate x with respect to t. Remember that a is treated as a constant. Using the power rule of differentiation (which states that the derivative of tⁿ is nt^(n-1)), we get:

dx/dt = 2at

This tells us how x changes with respect to t. It's the rate of change of x with respect to time, which makes sense because as t changes, x also changes. So we have our first piece of the puzzle.

Step 2: Find dy/dt

Next up, we have y = 2at. We need to differentiate y with respect to t to find dy/dt. Again, a is a constant. Applying the power rule (or just recognizing that the derivative of t with respect to itself is 1), we get:

dy/dt = 2a

This tells us how y changes with respect to t. It’s a constant rate, which means y changes linearly as t changes. Now we have our second piece of the puzzle.

Step 3: Calculate dy/dx

Now, here's the magic! We use the following formula, which is derived from the chain rule:

dy/dx = (dy/dt) / (dx/dt)

This is where we put everything together. Plugging in the values we found:

dy/dx = (2a) / (2at)

Step 4: Simplify

Let's simplify that expression: The 2a in the numerator and the 2at in the denominator both have a common factor of 2a. Dividing both the numerator and the denominator by 2a, we get:

dy/dx = 1/t

And there you have it! The derivative dy/dx = 1/t. This is the slope of the curve at any given point, defined by the parameter t. Notice that the slope changes depending on the value of t. This is typical for parametric equations; the slope isn't a constant, but rather a function of the parameter.

Conclusion: Understanding the Result

So, what does dy/dx = 1/t actually mean? It tells us the slope of the curve formed by the parametric equations x = at² and y = 2at at any point t. The slope changes as t changes. This means that the curve is not a straight line; it's a curve where the steepness varies. This is actually a parabola. For different values of t, we get different slopes. When t is large, the slope is shallow, and when t is close to zero, the slope becomes very steep (approaching infinity, but never actually touching because it is undefined at t = 0). This is the power of dy/dx: it gives us the instantaneous rate of change, showing us how y responds to x at any point along the curve.

Moreover, the process we followed is not just for this specific problem, but also a general method for solving many other parametric equations. The key takeaway is to grasp the chain rule and how to apply it, and to remember that finding the derivative involves breaking down the problem into smaller parts and using the chain rule to connect those parts. It involves understanding the concept of a parameter and how it influences the relationship between x and y. Keep practicing, and you'll become a pro at these problems! Remember, practice makes perfect. Keep working on different examples, and you'll build your skills and confidence. You got this, guys!