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The problem asks to find the partial derivative of the function \( z = x^3 + 7x^2y - 4xy^2 + y^3 + 2 \) with respect to \( y \) at the point \( A(1, -2) \).

### Steps:
1. **Find the partial derivative of \( z \) with respect to \( y \):**

\[
\frac{\partial z}{\partial y} = \frac{\partial}{\partial y} \left( x^3 + 7x^2y - 4xy^2 + y^3 + 2 \right)
\]

Taking the derivative term by term:
- \( \frac{\partial}{\partial y}(x^3) = 0 \) (since it does not depend on \( y \)),
- \( \frac{\partial}{\partial y}(7x^2y) = 7x^2 \),
- \( \frac{\partial}{\partial y}(-4xy^2) = -8xy \),
- \( \frac{\partial}{\partial y}(y^3) = 3y^2 \),
- \( \frac{\partial}{\partial y}(2) = 0 \) (since it is a constant).

Thus, the partial derivative of \( z \) with respect to \( y \) is:

\[
\frac{\partial z}{\partial y} = 7x^2 - 8xy + 3y^2
\]

2. **Substitute \( x = 1 \) and \( y = -2 \) into the partial derivative:**

\[
\frac{\partial z}{\partial y} \Big|_{(1, -2)} = 7(1)^2 - 8(1)(-2) + 3(-2)^2
\]
\[
= 7 + 16 + 12 = 35
\]

### Answer:
The value of the partial derivative of \( z \) with respect to \( y \) at point \( A(1, -2) \) is **35**.

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Would you like more details on any part of the solution? Here are 5 related questions to deepen your understanding:

1. How do you find the partial derivative of a function with respect to \( x \)?
2. What is the geometric interpretation of a partial derivative?
3. How can you apply partial derivatives to optimization problems?
4. How do mixed partial derivatives relate to each other (e.g., \( \frac{\partial^2 z}{\partial x \partial y} \))?
5. Can you solve partial derivatives for functions with more than two variables?

**Tip:** When calculating partial derivatives, treat all other variables as constants, and only focus on the variable you're differentiating with respect to.

Найти значение частной производной по y функции двух переменных z = x^3 + 7x^2y - 4xy^2 + y^3 + 2 в точке A(1, -2).

Find the partial derivative of the function z = x^3 + 7x^2y - 4xy^2 + y^3 + 2 with respect to y at the point A(1, -2). Detailed step-by-step solution using partial derivative formulas, suitable for university-level calculus students.

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