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To find the gradient of the function \( \Phi(x, y, z) = 3xz^2y - y^3z^2 \) at the point \( (1, -2, -1) \), we will follow these steps:

1. **Compute the partial derivatives** of \( \Phi \) with respect to each variable \( x \), \( y \), and \( z \).
2. **Evaluate the partial derivatives** at the point \( (1, -2, -1) \).
3. **Form the gradient vector** at that point.

### Step 1: Compute the Partial Derivatives
Let’s find each partial derivative:

1. **Partial derivative with respect to \( x \):**
   \[
   \frac{\partial \Phi}{\partial x} = 3z^2y
   \]

2. **Partial derivative with respect to \( y \):**
   \[
   \frac{\partial \Phi}{\partial y} = 3xz^2 - 3y^2z^2
   \]

3. **Partial derivative with respect to \( z \):**
   \[
   \frac{\partial \Phi}{\partial z} = 6xz y - 2y^3z
   \]

### Step 2: Evaluate the Partial Derivatives at \( (1, -2, -1) \)
Substituting \( x = 1 \), \( y = -2 \), and \( z = -1 \):

1. For \( \frac{\partial \Phi}{\partial x} \):
   \[
   \frac{\partial \Phi}{\partial x} = 3(-1)^2(-2) = 3 \cdot 1 \cdot (-2) = -6
   \]

2. For \( \frac{\partial \Phi}{\partial y} \):
   \[
   \frac{\partial \Phi}{\partial y} = 3 \cdot 1 \cdot (-1)^2 - 3(-2)^2(-1)^2
   \]
   \[
   = 3 \cdot 1 - 3 \cdot 4 \cdot 1 = 3 - 12 = -9
   \]

3. For \( \frac{\partial \Phi}{\partial z} \):
   \[
   \frac{\partial \Phi}{\partial z} = 6 \cdot 1 \cdot (-1) \cdot (-2) - 2(-2)^3(-1)
   \]
   \[
   = 12 - (-16) = 12 + 16 = 28
   \]

### Step 3: Form the Gradient Vector
The gradient vector at the point \( (1, -2, -1) \) is:
\[
\nabla \Phi = \left( \frac{\partial \Phi}{\partial x}, \frac{\partial \Phi}{\partial y}, \frac{\partial \Phi}{\partial z} \right) = (-6, -9, 28)
\]

### Final Answer
The gradient of \( \Phi \) at \( (1, -2, -1) \) is:
\[
\nabla \Phi = (-6, -9, 28)
\]

Would you like more details or have any questions?

### Related Questions
1. What is the physical interpretation of the gradient vector?
2. How do you determine the direction of the steepest ascent from a gradient?
3. What happens to the gradient if you change the point of evaluation?
4. How do partial derivatives change when you modify one of the variables in \( \Phi \)?
5. Can you find the directional derivative of \( \Phi \) at \( (1, -2, -1) \) in a given direction?

### Tip
The gradient points in the direction of the steepest increase of the function, and its magnitude indicates how steep the slope is in that direction.

If Φ(x, y, z) = 3xz^2y - y^3z^2, find Grad Φ at point (1, -2, -1).

Learn how to find the gradient of the multivariable function Φ(x, y, z) = 3xz^2y - y^3z^2 at the point (1, -2, -1). This solution covers the calculation of partial derivatives and assembling the gradient vector. Suitable for college-level calculus students.

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