We are asked to find \(\frac{dy}{dx}\) for the equation:

\[
2x^3 - 5xy - y^2 = 3
\]

Since this equation involves both \(x\) and \(y\), we need to use **implicit differentiation**. Let's proceed step by step.

### Step 1: Differentiate each term with respect to \(x\)

1. \( \frac{d}{dx}(2x^3) = 6x^2 \)
2. \( \frac{d}{dx}(-5xy) \) — This requires the product rule:
   \[
   \frac{d}{dx}(-5xy) = -5\left(\frac{d}{dx}(x \cdot y)\right) = -5\left(1 \cdot y + x \cdot \frac{dy}{dx}\right) = -5y - 5x \frac{dy}{dx}
   \]
3. \( \frac{d}{dx}(-y^2) = -2y \frac{dy}{dx} \) — Since \(y\) is a function of \(x\), we apply the chain rule.

### Step 2: Differentiate the right-hand side
\[
\frac{d}{dx}(3) = 0
\]

### Step 3: Combine the results
After differentiating each term, we get:

\[
6x^2 - 5y - 5x \frac{dy}{dx} - 2y \frac{dy}{dx} = 0
\]

### Step 4: Solve for \(\frac{dy}{dx}\)
Now, let's solve for \(\frac{dy}{dx}\):

\[
6x^2 - 5y = 5x \frac{dy}{dx} + 2y \frac{dy}{dx}
\]

Factor out \(\frac{dy}{dx}\) on the right-hand side:

\[
6x^2 - 5y = \left(5x + 2y\right) \frac{dy}{dx}
\]

Finally, divide both sides by \(5x + 2y\) to isolate \(\frac{dy}{dx}\):

\[
\frac{dy}{dx} = \frac{6x^2 - 5y}{5x + 2y}
\]

This is the derivative \(\frac{dy}{dx}\).

---

Would you like more details on any part of this process or have any further questions?

Here are 5 related questions to expand on this topic:

1. What is the product rule, and how does it apply in implicit differentiation?
2. Can you show how to solve another implicit differentiation problem with a trigonometric function?
3. How do you apply the chain rule when differentiating composite functions?
4. Why do we need implicit differentiation when differentiating functions involving both \(x\) and \(y\)?
5. How would the solution change if the original equation was \(2x^3 - 5xy + y^2 = 3\)?

**Tip:** When differentiating terms like \(xy\), always apply the product rule, since both \(x\) and \(y\) are functions of \(x\).

Discover how to solve the implicit differentiation problem for 2x^3 - 5xy - y^2 = 3 with a step-by-step explanation. Learn about key calculus concepts such as the product rule and chain rule, and how to apply them to find dy/dx. Suitable for Grades 11-12 or early college students.

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