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To find \(\frac{dy}{dx}\) for the equation

\[
y^{10} e^{4x} - 3x^7 - y^2 = 1,
\]

we'll differentiate both sides with respect to \(x\) using **implicit differentiation**.

### Step 1: Differentiate each term
1. **Differentiate \(y^{10} e^{4x}\):**
   - Use the product rule: \(\frac{d}{dx} \left( y^{10} e^{4x} \right) = \frac{d}{dx}(y^{10}) \cdot e^{4x} + y^{10} \cdot \frac{d}{dx}(e^{4x})\).
   - For \(\frac{d}{dx}(y^{10})\): Apply the chain rule to get \(10y^9 \frac{dy}{dx}\).
   - For \(\frac{d}{dx}(e^{4x})\): This derivative is \(4e^{4x}\).

   Combining these, we have:
   \[
   \frac{d}{dx} \left( y^{10} e^{4x} \right) = 10y^9 \frac{dy}{dx} e^{4x} + 4y^{10} e^{4x}.
   \]

2. **Differentiate \(-3x^7\):**
   - This term differentiates as \(-21x^6\).

3. **Differentiate \(-y^2\):**
   - Using the chain rule, this becomes \(-2y \frac{dy}{dx}\).

4. **Differentiate the constant \(1\):**
   - The derivative of a constant is \(0\).

### Step 2: Combine the derivatives
Now, rewrite the differentiated equation:

\[
10y^9 \frac{dy}{dx} e^{4x} + 4y^{10} e^{4x} - 21x^6 - 2y \frac{dy}{dx} = 0.
\]

### Step 3: Solve for \(\frac{dy}{dx}\)
Combine terms that include \(\frac{dy}{dx}\):

\[
\left(10y^9 e^{4x} - 2y\right) \frac{dy}{dx} = -4y^{10} e^{4x} + 21x^6.
\]

Now, isolate \(\frac{dy}{dx}\):

\[
\frac{dy}{dx} = \frac{-4y^{10} e^{4x} + 21x^6}{10y^9 e^{4x} - 2y}.
\]

### Final Answer:
\[
\frac{dy}{dx} = \frac{-4y^{10} e^{4x} + 21x^6}{10y^9 e^{4x} - 2y}.
\]

Would you like more details on any part of this solution?

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Here are five related questions to expand your understanding:

1. How would the differentiation change if there was an additional term, such as \(x^3y\), on the left side of the equation?
2. What would happen if the exponent on \(y\) was different, say \(y^5\), instead of \(y^{10}\)?
3. How can implicit differentiation be used to find the second derivative, \(\frac{d^2y}{dx^2}\)?
4. Why does the product rule apply to terms like \(y^{10} e^{4x}\)?
5. How does implicit differentiation differ from explicit differentiation?

**Tip:** Always check each step of implicit differentiation carefully, especially when differentiating terms involving both \(x\) and \(y\).

Find \( \frac{dy}{dx} \) where \( y^{10} e^{4x} - 3x^7 - y^2 = 1 \).

Learn how to find \( \frac{dy}{dx} \) for the equation \( y^{10} e^{4x} - 3x^7 - y^2 = 1 \) using implicit differentiation. This problem covers the product rule, chain rule, and implicit differentiation techniques, suitable for advanced high school or early college students.

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