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To solve the problem using implicit differentiation, we are given the equation:

\[
x^6 w + w^7 + w z^2 + 5yz = 0
\]

We need to find \(\frac{\partial w}{\partial z}\). To achieve this, we will differentiate both sides of the equation with respect to \(z\), treating \(w\) as a function of \(z\).

### Steps for implicit differentiation:

1. Differentiate \(x^6w\) with respect to \(z\):
   \[
   \frac{\partial}{\partial z} (x^6 w) = x^6 \frac{\partial w}{\partial z} = x^6 \frac{dw}{dz}
   \]
   Since \(x\) is not a function of \(z\), we treat it as a constant.

2. Differentiate \(w^7\) with respect to \(z\):
   \[
   \frac{\partial}{\partial z} (w^7) = 7w^6 \frac{\partial w}{\partial z} = 7w^6 \frac{dw}{dz}
   \]

3. Differentiate \(wz^2\) with respect to \(z\):
   Here, we need to use the product rule:
   \[
   \frac{\partial}{\partial z} (wz^2) = z^2 \frac{dw}{dz} + w \cdot 2z
   \]

4. Differentiate \(5yz\) with respect to \(z\):
   \[
   \frac{\partial}{\partial z} (5yz) = 5y
   \]
   Again, \(y\) is not a function of \(z\), so it is treated as a constant.

Now, putting everything together:

\[
x^6 \frac{dw}{dz} + 7w^6 \frac{dw}{dz} + z^2 \frac{dw}{dz} + 2wz + 5y = 0
\]

Now, we solve for \(\frac{dw}{dz}\):

\[
\frac{dw}{dz}(x^6 + 7w^6 + z^2) = - (2wz + 5y)
\]

Thus, the derivative is:

\[
\frac{dw}{dz} = \frac{-(2wz + 5y)}{x^6 + 7w^6 + z^2}
\]

This is the expression for \(\frac{dw}{dz}\).

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Would you like further details or clarifications? Here are five related questions:

1. How is implicit differentiation different from explicit differentiation?
2. Can this method be extended to higher-order derivatives?
3. What is the significance of treating some variables as constants in implicit differentiation?
4. How does the product rule apply in cases like \(wz^2\)?
5. Can implicit differentiation be used for non-polynomial functions?

**Tip:** Always remember to apply the chain rule when differentiating implicitly, especially when dealing with products of variables.

Calculate the derivative using implicit differentiation: ∂w/∂z for the equation x^6w + w^7 + wz^2 + 5yz = 0.

Learn how to use implicit differentiation to find ∂w/∂z for the equation x^6w + w^7 + wz^2 + 5yz = 0. This solution involves applying the chain rule and product rule. Suitable for undergraduate-level calculus students.

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