https://file.aiquickdraw.com/mathbot/file/1725442660622dx7n52zp.jpg

The expression in the image is:

\[
\phi(x, y, z) = 3x^2y - y^2z
\]

We are asked to find the gradient of \(\phi\) (denoted as \(\nabla \phi\)) at the point \((1, 2, 3)\).

### Step 1: Calculate the Partial Derivatives
The gradient \(\nabla \phi\) is a vector of partial derivatives of \(\phi\) with respect to \(x\), \(y\), and \(z\):

\[
\nabla \phi = \left( \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z} \right)
\]

#### Partial Derivative with respect to \(x\):
\[
\frac{\partial \phi}{\partial x} = \frac{\partial}{\partial x}(3x^2y - y^2z) = 6xy
\]

#### Partial Derivative with respect to \(y\):
\[
\frac{\partial \phi}{\partial y} = \frac{\partial}{\partial y}(3x^2y - y^2z) = 3x^2 - 2yz
\]

#### Partial Derivative with respect to \(z\):
\[
\frac{\partial \phi}{\partial z} = \frac{\partial}{\partial z}(3x^2y - y^2z) = -y^2
\]

### Step 2: Evaluate the Gradient at the Point \((1, 2, 3)\)
Substitute \(x = 1\), \(y = 2\), and \(z = 3\) into the partial derivatives:

- \(\frac{\partial \phi}{\partial x} = 6xy = 6(1)(2) = 12\)
- \(\frac{\partial \phi}{\partial y} = 3x^2 - 2yz = 3(1)^2 - 2(2)(3) = 3 - 12 = -9\)
- \(\frac{\partial \phi}{\partial z} = -y^2 = -(2)^2 = -4\)

### Final Answer: The Gradient at \((1, 2, 3)\)
\[
\nabla \phi(1, 2, 3) = \langle 12, -9, -4 \rangle
\]

Would you like more details or have any questions?

### Related Questions:
1. How is the gradient of a function related to the direction of steepest ascent?
2. Can you explain why the gradient is a vector and not a scalar?
3. How would you find the gradient of a function with more than three variables?
4. What is the physical interpretation of the gradient in a 3D space?
5. How would the gradient change if the function \(\phi(x, y, z)\) included trigonometric terms?

### Tip:
When computing gradients, remember that each partial derivative represents the rate of change of the function with respect to one variable, holding the others constant.

If φ(x, y, z) = 3x²y - y²z, then grad φ at the point (1, 2, 3) is?

Learn how to calculate the gradient of the multivariable function φ(x, y, z) = 3x²y - y²z at the point (1, 2, 3). This problem involves key concepts in multivariable calculus, including the computation of partial derivatives and the application of the gradient formula. Suitable for undergraduate-level students.

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