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The question involves finding the partial derivative \(\frac{\partial z}{\partial x}\) based on the given implicit equation:

\[
x^2 yz + y^2 x^3 + xy = x^2
\]

We will solve this using implicit differentiation with respect to \(x\), treating \(z\) as an implicit function of both \(x\) and \(y\).

### Step-by-Step Solution:

1. **Original Equation:**
   \[
   x^2 yz + y^2 x^3 + xy = x^2
   \]
   
2. **Differentiate each term with respect to \(x\):**

   - For \(x^2 yz\), apply the product rule:
     \[
     \frac{d}{dx}(x^2 yz) = 2x yz + x^2 y \frac{\partial z}{\partial x}
     \]

   - For \(y^2 x^3\):
     \[
     \frac{d}{dx}(y^2 x^3) = 3x^2 y^2
     \]

   - For \(xy\), apply the product rule:
     \[
     \frac{d}{dx}(xy) = y + x \frac{dy}{dx}
     \]
   
   - For \(x^2\):
     \[
     \frac{d}{dx}(x^2) = 2x
     \]
   
3. **Combine all the differentiated parts:**
   \[
   2x yz + x^2 y \frac{\partial z}{\partial x} + 3x^2 y^2 + y = 2x
   \]

4. **Solve for \(\frac{\partial z}{\partial x}\):**
   Rearranging the equation to isolate \(\frac{\partial z}{\partial x}\):
   \[
   x^2 y \frac{\partial z}{\partial x} = 2x - 2x yz - 3x^2 y^2 - y
   \]
   \[
   \frac{\partial z}{\partial x} = \frac{2x - 2x yz - 3x^2 y^2 - y}{x^2 y}
   \]

Now, compare this result with the given options in the image.

The correct answer is the first option:

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

### Would you like further clarification on any part of the solution?

#### Here are 5 related questions you may explore:
1. How do you compute implicit differentiation for multivariable functions?
2. How can you apply the chain rule in implicit differentiation?
3. What is the significance of partial derivatives in implicit functions?
4. How does the product rule apply when differentiating terms with multiple variables?
5. Can we find \(\frac{\partial z}{\partial y}\) for the same equation?

**Tip**: Always double-check each derivative step by applying the product and chain rules properly when differentiating implicitly.

Cho z là hàm ẩn theo biến x và y được xác định bởi phương trình x^2yz + y^2x^3 + xy = x^2. Tìm đạo hàm riêng ∂z/∂x.

Learn how to find the partial derivative of z with respect to x using implicit differentiation for the equation x²yz + y²x³ + xy = x². This solution involves using the product rule and isolating ∂z/∂x.

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