Solving For Dy/dx: A Step-by-Step Guide

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Hey guys! Let's dive into a classic calculus problem where we need to find the derivative dy/dx given an implicit equation. This is super common in math, and understanding the process is key. We're going to break down how to solve for dy/dx when you're given the equation $3x^2 + 5x2y2 = 2y$. Don't worry if it looks intimidating at first; we'll take it one step at a time, making sure we get this down. Implicit differentiation is all about treating y as a function of x, even if it's not explicitly defined. This means we'll need to use the chain rule quite a bit. Ready? Let's get started!

Understanding Implicit Differentiation and the Chain Rule

First off, let's make sure we're all on the same page with implicit differentiation. Unlike regular differentiation, where you have a function like y = f(x), implicit differentiation deals with equations where x and y are mixed together, and it's not easy to isolate y. Think of it like this: y is lurking in the shadows, depending on x in a way that's not immediately obvious. The chain rule is our best friend here. If we have something like y², we treat it as (y(x))², and when we differentiate, we have to consider the derivative of y with respect to x, which is dy/dx. So, if we differentiate y², we get 2y(dy/dx)*. This is because we're taking the derivative of the 'outside' function (the square), and then multiplying it by the derivative of the 'inside' function (y), which is dy/dx. The trick is to remember that every time you differentiate a term with y, you'll need to tack on a dy/dx.

Core Concepts

  • Implicit Differentiation: Differentiating equations where x and y are mixed together.
  • Chain Rule: Essential for differentiating terms involving y. If we differentiate a term with y (e.g., y²), we treat it as (y(x))² and include dy/dx.

Now, let's get down to the actual calculation. Keep these concepts in mind, and you'll be golden. Let's apply this to our given equation: $3x^2 + 5x2y2 = 2y$. We'll differentiate both sides with respect to x.

Step-by-Step Differentiation of the Equation

Alright, let's get our hands dirty and differentiate the equation $3x^2 + 5x2y2 = 2y$. We'll go term by term. For the term $3x^2$, the derivative is simply $6x$. Easy peasy! Now, for the term $5x2y2$, we're going to need the product rule and the chain rule. Remember, the product rule states that the derivative of uv is u'v + uv'. Here, u = 5x² and v = y². So, we differentiate each part separately. The derivative of $5x^2$ is $10x$, and the derivative of $y^2$ with respect to x is $2y*(dy/dx)$, applying the chain rule. Thus, the derivative of $5x2y2$ becomes $10xy^2 + 5x^2(2y)(dy/dx) = 10xy^2 + 10x^2y(dy/dx)$. Lastly, the derivative of $2y$ with respect to x is $2(dy/dx)$. So, we have now differentiated every term in our original equation. This is the heart of the process; now, we'll organize it. Always remember to break it down piece by piece.

Detailed Breakdown

  1. Differentiate $3x^2$: Resulting in $6x$.
  2. Differentiate $5x2y2$: Applying the product rule and chain rule results in $10xy^2 + 10x^2y(dy/dx)$.
  3. Differentiate $2y$: Applying the chain rule results in $2(dy/dx)$.

Keep everything organized, and you'll do great. Now, we'll combine all these derivatives and solve for $dy/dx$.

Solving for dy/dx

Okay, guys, we've done the hard part – differentiating! Now, we're going to combine all the derivatives and solve for dy/dx. Remember, we differentiated both sides of the equation. So, we'll put everything together. The derivative of $3x^2 + 5x2y2 = 2y$ becomes: $6x + 10xy^2 + 10x^2y(dy/dx) = 2(dy/dx)$. Now we want to isolate dy/dx. First, move all the terms containing dy/dx to one side and the other terms to the other side. This gives us: $10x^2y(dy/dx) - 2(dy/dx) = -6x - 10xy^2$. Now, factor out dy/dx: $dy/dx(10x^2y - 2) = -6x - 10xy^2$. Finally, divide both sides by $(10x^2y - 2)$ to solve for dy/dx: $dy/dx = (-6x - 10xy^2) / (10x^2y - 2)$. And there you have it! We've successfully found dy/dx.

The Final Steps

  1. Combine the derivatives: $6x + 10xy^2 + 10x^2y(dy/dx) = 2(dy/dx)$.
  2. Isolate dy/dx: $10x^2y(dy/dx) - 2(dy/dx) = -6x - 10xy^2$.
  3. Factor out dy/dx: $dy/dx(10x^2y - 2) = -6x - 10xy^2$.
  4. Solve for dy/dx: $dy/dx = (-6x - 10xy^2) / (10x^2y - 2)$.

You've done it! Let's wrap things up and look at some key takeaways.

Conclusion and Key Takeaways

Congrats, we've successfully found dy/dx for the equation $3x^2 + 5x2y2 = 2y$! The final answer is: $dy/dx = (-6x - 10xy^2) / (10x^2y - 2)$. The main idea here is to remember the chain rule when differentiating terms with y and to use the product rule when necessary. Implicit differentiation is a powerful technique, and it's essential for a lot of advanced calculus problems. Practice makes perfect, so keep working through problems like this one. You'll get more comfortable with it, and it will become second nature! Remember to break down each step and always double-check your work, especially when applying the chain rule. You got this, folks!

Final Points to Remember

  • Master the chain rule for terms involving y.
  • Understand and correctly apply the product rule when necessary.
  • Practice, practice, practice! The more problems you solve, the more comfortable you'll become.

Keep up the great work, and happy calculating! If you enjoyed this, check out more tutorials. Keep learning, keep growing, and don't be afraid to tackle tough problems. You've got the skills now to approach these types of problems with confidence.