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To find \(\frac{dy}{dx}\) using implicit differentiation for the equation:

\[
\cos(3y^2) + 8x = 8e^y
\]

we'll differentiate both sides with respect to \(x\), treating \(y\) as a function of \(x\).

### Step-by-Step Solution:

1. **Differentiate the Left Side:**

   \[
   \frac{d}{dx}\left(\cos(3y^2) + 8x\right)
   \]

   - For \(\cos(3y^2)\):
     \[
     \frac{d}{dx} \cos(3y^2) = -\sin(3y^2) \cdot \frac{d}{dx}(3y^2)
     \]
     - Now, differentiate \(3y^2\) with respect to \(x\):
       \[
       \frac{d}{dx}(3y^2) = 6y \cdot \frac{dy}{dx}
       \]
     - So,
       \[
       \frac{d}{dx} \cos(3y^2) = -\sin(3y^2) \cdot 6y \cdot \frac{dy}{dx}
       \]

   - For \(8x\):
     \[
     \frac{d}{dx}(8x) = 8
     \]

   - Putting it together, the left side derivative is:
     \[
     -6y \sin(3y^2) \cdot \frac{dy}{dx} + 8
     \]

2. **Differentiate the Right Side:**

   \[
   \frac{d}{dx}(8e^y)
   \]
   - For \(8e^y\):
     \[
     \frac{d}{dx}(8e^y) = 8e^y \cdot \frac{dy}{dx}
     \]

3. **Set Up the Equation:**

   After differentiating both sides, we have:
   \[
   -6y \sin(3y^2) \cdot \frac{dy}{dx} + 8 = 8e^y \cdot \frac{dy}{dx}
   \]

4. **Solve for \(\frac{dy}{dx}\):**

   - Move terms with \(\frac{dy}{dx}\) to one side:
     \[
     8 = 8e^y \cdot \frac{dy}{dx} + 6y \sin(3y^2) \cdot \frac{dy}{dx}
     \]
   - Factor out \(\frac{dy}{dx}\):
     \[
     8 = \left(8e^y + 6y \sin(3y^2)\right) \cdot \frac{dy}{dx}
     \]
   - Divide by \(8e^y + 6y \sin(3y^2)\):
     \[
     \frac{dy}{dx} = \frac{8}{8e^y + 6y \sin(3y^2)}
     \]

### Final Answer:

\[
\frac{dy}{dx} = \frac{8}{8e^y + 6y \sin(3y^2)}
\]

Let me know if you want further clarification on any steps, or if you'd like more details on implicit differentiation!

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Here are some related questions to deepen your understanding:

1. How would the solution change if we had \(\sin(3y^2)\) instead of \(\cos(3y^2)\) in the original equation?
2. What is the significance of treating \(y\) as a function of \(x\) in implicit differentiation?
3. Could you explain why the chain rule is essential in implicit differentiation?
4. How would you find the second derivative \(\frac{d^2y}{dx^2}\) for this problem?
5. How would implicit differentiation work if the equation involved higher powers of \(y\)?

**Tip:** Always apply the chain rule carefully when differentiating terms with \(y\), as it often involves both \(y\) and \(\frac{dy}{dx}\).

Use implicit differentiation to find dy/dx for cos(3y^2) + 8x = 8e^y.

Learn how to use implicit differentiation to find dy/dx for the equation cos(3y^2) + 8x = 8e^y. This step-by-step solution applies the chain rule and differentiates trigonometric and exponential functions with respect to x. Suitable for college calculus students.

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