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We are tasked with finding \( \frac{\partial z}{\partial x} \) at the point \( (1, 1, 1) \), given the implicit equation:

\[
xy + z^3x - 2yz = 0
\]

We will use implicit differentiation to solve this problem. Here's the step-by-step process:

### Step 1: Differentiate implicitly with respect to \( x \)
- Remember, \( z \) is a function of both \( x \) and \( y \), so when differentiating any term involving \( z \), we must apply the chain rule.

#### Differentiate each term:

1. \( \frac{d}{dx}(xy) = y + x \frac{\partial y}{\partial x} \) but since \( y \) is treated as independent of \( x \), this simplifies to \( y \).

2. \( \frac{d}{dx}(z^3x) = z^3 + 3z^2x \frac{\partial z}{\partial x} \) (product rule applied, chain rule on \( z^3 \)).

3. \( \frac{d}{dx}(-2yz) = -2y \frac{\partial z}{\partial x} \) (since \( y \) is independent of \( x \), we only differentiate \( z \)).

Now, putting these together:

\[
y + z^3 + 3z^2x \frac{\partial z}{\partial x} - 2y \frac{\partial z}{\partial x} = 0
\]

### Step 2: Solve for \( \frac{\partial z}{\partial x} \)
Group the terms containing \( \frac{\partial z}{\partial x} \) on one side:

\[
3z^2x \frac{\partial z}{\partial x} - 2y \frac{\partial z}{\partial x} = -y - z^3
\]

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

\[
\frac{\partial z}{\partial x} (3z^2x - 2y) = -y - z^3
\]

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

\[
\frac{\partial z}{\partial x} = \frac{-y - z^3}{3z^2x - 2y}
\]

### Step 3: Substitute \( x = 1, y = 1, z = 1 \)

Substitute the values \( x = 1 \), \( y = 1 \), and \( z = 1 \) into the equation:

\[
\frac{\partial z}{\partial x} = \frac{-(1) - 1^3}{3(1^2)(1) - 2(1)} = \frac{-1 - 1}{3 - 2} = \frac{-2}{1} = -2
\]

Thus, the value of \( \frac{\partial z}{\partial x} \) at the point \( (1, 1, 1) \) is \( -2 \).

Would you like any further details or clarifications?

Here are some follow-up questions for further exploration:
1. How does implicit differentiation work when the function is dependent on multiple variables?
2. What is the geometric interpretation of partial derivatives in 3D space?
3. How can we compute \( \frac{\partial z}{\partial y} \) using a similar method?
4. What would change in the solution if \( y \) were also dependent on \( x \)?
5. How does the chain rule apply in implicit differentiation?

**Tip:** When working with implicit differentiation, always be careful to apply the chain rule when differentiating variables that depend on the one you're differentiating with respect to.

Find the value of ∂z/∂x at the point (1, 1, 1) if the equation xy + z^3x - 2yz = 0 defines z as a function of the two independent variables x and y, and the partial derivative exists.

Learn how to find the partial derivative ∂z/∂x using implicit differentiation for the equation xy + z^3x - 2yz = 0 at the point (1, 1, 1). This problem involves key concepts like the chain rule and implicit differentiation, suited for university-level calculus.

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