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To solve for \( F_y \), we need to find the partial derivative of the given function \( F(x, y) \) with respect to \( y \).

The equation is:

\[
F(x, y) = \sqrt{x^2 + 6xy + y^6} - 8 = 0
\]

We are tasked with calculating \( F_y \), which is the partial derivative of \( F(x, y) \) with respect to \( y \).

Let's proceed step-by-step:

1. The first part is the square root term: \( \sqrt{x^2 + 6xy + y^6} \). We apply the chain rule to differentiate it with respect to \( y \).
   
2. Then, we compute the derivative of the expression inside the square root: \( x^2 + 6xy + y^6 \).

Let me calculate \( F_y \).The partial derivative of \( F(x, y) \) with respect to \( y \), denoted \( F_y \), is:

\[
F_y = \frac{3(x + y^5)}{\sqrt{x^2 + 6xy + y^6}}
\]

Would you like further explanation on how this derivative was computed or have any other questions?

Here are 5 related questions to explore:
1. How do you compute partial derivatives for multivariable functions?
2. What is the chain rule, and how is it applied in partial differentiation?
3. What are the geometric interpretations of partial derivatives?
4. How does the behavior of \( F(x, y) \) change with respect to \( x \) and \( y \)?
5. Can this expression for \( F_y \) be simplified further under certain conditions?

**Tip:** Always ensure that you apply the chain rule properly when dealing with composite functions, especially in multivariable calculus.

If F(x,y) = sqrt(x^2 + 6xy + y^6) - 8 = 0, then Fy = ?

This problem involves finding the partial derivative Fy of the function F(x, y) = sqrt(x^2 + 6xy + y^6) - 8 = 0 using the chain rule. The final expression for Fy is (3(x + y^5)) / sqrt(x^2 + 6xy + y^6). Suitable for undergraduate students studying multivariable calculus.

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