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The problem asks us to find \( \frac{\partial z}{\partial y} \) for the implicit function defined by the equation:

\[
xy + yz + xz + \sin(xyz) = 3
\]

We are given the point \( (x, y, z) = (3, 1, 0) \). Let's go through the steps to solve this.

### Step 1: Implicit Differentiation

We will differentiate both sides of the equation with respect to \( y \), treating \( z \) as a function of \( y \). Using implicit differentiation:

\[
\frac{d}{dy}\left(xy + yz + xz + \sin(xyz)\right) = \frac{d}{dy}(3)
\]

Since the right-hand side is a constant, its derivative is 0.

Now differentiate each term on the left-hand side with respect to \( y \):

- \( \frac{d}{dy}(xy) = x \)
- \( \frac{d}{dy}(yz) = z + y \frac{\partial z}{\partial y} \)
- \( \frac{d}{dy}(xz) = x \frac{\partial z}{\partial y} \)
- \( \frac{d}{dy}(\sin(xyz)) = \cos(xyz) \cdot \frac{d}{dy}(xyz) = \cos(xyz) \cdot (xz + xy \frac{\partial z}{\partial y}) \)

### Step 2: Substitute the Point \( (x, y, z) = (3, 1, 0) \)

Next, we substitute \( x = 3 \), \( y = 1 \), and \( z = 0 \) into the equation:

- \( x = 3 \)
- \( y = 1 \)
- \( z = 0 \)

Differentiating each term:

- \( \frac{d}{dy}(xy) = 3 \)
- \( \frac{d}{dy}(yz) = 0 + 1 \cdot \frac{\partial z}{\partial y} = \frac{\partial z}{\partial y} \)
- \( \frac{d}{dy}(xz) = 3 \cdot \frac{\partial z}{\partial y} = 3 \frac{\partial z}{\partial y} \)
- \( \frac{d}{dy}(\sin(xyz)) = \cos(0) \cdot (3 \cdot 0 + 3 \cdot 1 \cdot \frac{\partial z}{\partial y}) = 1 \cdot 3 \frac{\partial z}{\partial y} = 3 \frac{\partial z}{\partial y} \)

### Step 3: Set Up the Equation

Now, we set up the equation:

\[
3 + \frac{\partial z}{\partial y} + 3 \frac{\partial z}{\partial y} + 3 \frac{\partial z}{\partial y} = 0
\]

Simplifying:

\[
3 + 7 \frac{\partial z}{\partial y} = 0
\]

### Step 4: Solve for \( \frac{\partial z}{\partial y} \)

Now, solve for \( \frac{\partial z}{\partial y} \):

\[
7 \frac{\partial z}{\partial y} = -3
\]
\[
\frac{\partial z}{\partial y} = -\frac{3}{7}
\]

### Final Answer:
The value of \( \frac{\partial z}{\partial y} \) at the point \( (3, 1, 0) \) is \( -\frac{3}{7} \).

Would you like further details, or do you have any other questions?

### Relative Questions:
1. How would you approach the problem if you were asked to find \( \frac{\partial z}{\partial x} \) instead?
2. What changes would occur if the equation involved a different trigonometric function, such as \( \cos(xyz) \)?
3. How does implicit differentiation work in multivariable calculus?
4. What if you were asked to find \( \frac{\partial z}{\partial y} \) at a different point, such as \( (2, 1, 1) \)?
5. How can you interpret the result of \( \frac{\partial z}{\partial y} \) geometrically?

### Tip:
When differentiating implicitly with respect to one variable in multivariable functions, always treat other variables as constants unless they depend on the variable you're differentiating with respect to.

If xy + yz + xz + sin(xyz) = 3, find ∂z/∂y at (3, 1, 0).

Learn how to use implicit differentiation to find the partial derivative ∂z/∂y for the equation xy + yz + xz + sin(xyz) = 3 at the point (3, 1, 0). This problem covers multivariable calculus concepts and applies the implicit function theorem.

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