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?

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 $y$.