Use implicit differentiation to calculate the derivative 
∂z
∂x
 if 
xyz + x2y + y4z = 0.

We are given the equation:

\[
xyz + x^2y + y^4z = 0
\]

We need to find \(\frac{\partial z}{\partial x}\) using implicit differentiation. Here's the step-by-step process.

### Step 1: Differentiate both sides of the equation with respect to \(x\).
Since \(z = z(x, y)\), we must apply the product rule when differentiating terms involving \(z\).

1. **First term: \(xyz\)**:
   Using the product rule:
   \[
   \frac{d}{dx}(xyz) = \frac{d}{dx}(x) \cdot yz + x \cdot \frac{d}{dx}(yz)
   \]
   - The first part is \(yz\), since \(\frac{d}{dx}(x) = 1\).
   - For the second part, use the product rule again:
     \[
     \frac{d}{dx}(yz) = y \cdot \frac{d}{dx}(z) = y \cdot \frac{\partial z}{\partial x}
     \]
   So, the derivative of \(xyz\) becomes:
   \[
   yz + xy \frac{\partial z}{\partial x}
   \]

2. **Second term: \(x^2y\)**:
   Again, apply the product rule:
   \[
   \frac{d}{dx}(x^2y) = \frac{d}{dx}(x^2) \cdot y + x^2 \cdot \frac{d}{dx}(y)
   \]
   Since \(y\) is independent of \(x\), this simplifies to:
   \[
   2xy
   \]

3. **Third term: \(y^4z\)**:
   Here, only \(z\) depends on \(x\), so we use the chain rule:
   \[
   \frac{d}{dx}(y^4z) = y^4 \frac{\partial z}{\partial x}
   \]

### Step 2: Differentiate the right-hand side (constant \(0\)).

\[
\frac{d}{dx}(0) = 0
\]

### Step 3: Combine the results.

Putting everything together:

\[
yz + xy \frac{\partial z}{\partial x} + 2xy + y^4 \frac{\partial z}{\partial x} = 0
\]

### Step 4: Solve for \(\frac{\partial z}{\partial x}\).

Group the terms involving \(\frac{\partial z}{\partial x}\):

\[
xy \frac{\partial z}{\partial x} + y^4 \frac{\partial z}{\partial x} = -yz - 2xy
\]

Factor out \(\frac{\partial z}{\partial x}\):

\[
\frac{\partial z}{\partial x}(xy + y^4) = -yz - 2xy
\]

Finally, solve for \(\frac{\partial z}{\partial x}\):

\[
\frac{\partial z}{\partial x} = \frac{-yz - 2xy}{xy + y^4}
\]

This is the derivative \(\frac{\partial z}{\partial x}\) using implicit differentiation.

---

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

Here are 5 related questions for further exploration:
1. How would you calculate \(\frac{\partial z}{\partial y}\) for the same equation?
2. Can you explain the product rule in more detail?
3. What happens if we substitute specific values for \(x\), \(y\), and \(z\) into this result?
4. How does implicit differentiation differ from regular differentiation?
5. What if the equation involved more complex functions like trigonometric terms?

**Tip**: Always apply the product rule carefully when differentiating terms that involve products of functions of different variables.

Use implicit differentiation to calculate the derivative ∂z/∂x if xyz + x^2y + y^4z = 0.

Explore how to use implicit differentiation to find the partial derivative ∂z/∂x for the equation xyz + x^2y + y^4z = 0. This problem covers key concepts in multivariable calculus, such as the product rule and chain rule. Suitable for college-level calculus students.

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