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We are tasked with finding \( f'_y \) (the partial derivative of \( f \) with respect to \( y \)) at the point \( M_0 = (2, 0, 4) \) for the function 

\[
f(x, y, z) = e^{xyz} + (x+1)^3(y+1)^2 + \sin(z-4).
\]

### Step 1: Compute \( f'_y \)
The function is:

\[
f(x, y, z) = e^{xyz} + (x+1)^3(y+1)^2 + \sin(z-4).
\]

#### First term: \( e^{xyz} \)
The partial derivative with respect to \( y \) is:

\[
\frac{\partial}{\partial y} \left( e^{xyz} \right) = xz \cdot e^{xyz}.
\]

#### Second term: \( (x+1)^3(y+1)^2 \)
The partial derivative with respect to \( y \) is:

\[
\frac{\partial}{\partial y} \left( (x+1)^3(y+1)^2 \right) = 2(x+1)^3(y+1).
\]

#### Third term: \( \sin(z-4) \)
This term does not depend on \( y \), so its derivative with respect to \( y \) is:

\[
\frac{\partial}{\partial y} \left( \sin(z-4) \right) = 0.
\]

Thus, the partial derivative \( f'_y \) is:

\[
f'_y = xz \cdot e^{xyz} + 2(x+1)^3(y+1).
\]

---

### Step 2: Evaluate \( f'_y \) at \( M_0 = (2, 0, 4) \)
At the point \( M_0 = (2, 0, 4) \), substitute \( x = 2 \), \( y = 0 \), and \( z = 4 \) into \( f'_y \):

#### First term:
\[
xz \cdot e^{xyz} = 2 \cdot 4 \cdot e^{2 \cdot 0 \cdot 4} = 8 \cdot e^0 = 8.
\]

#### Second term:
\[
2(x+1)^3(y+1) = 2(2+1)^3(0+1) = 2 \cdot 3^3 \cdot 1 = 2 \cdot 27 = 54.
\]

#### Third term:
This term is 0, as calculated earlier.

Add the results:

\[
f'_y = 8 + 54 = 62.
\]

---

### Final Answer:
The correct answer is \( \mathbf{v)} \, 62 \).

---

Would you like a more detailed explanation, or do you have any specific questions?

### Related Questions:
1. What are the partial derivatives \( f'_x \) and \( f'_z \) for the same function at \( M_0 \)?
2. How would the calculation change if the function contained \( \cos(yz) \) instead of \( \sin(z-4) \)?
3. How is the gradient of a multivariable function computed, and how is it related to partial derivatives?
4. What is the significance of partial derivatives in optimization problems?
5. Can you explain the chain rule in the context of multivariable calculus?

### Tip:
When evaluating derivatives, always check if the variables in the terms are independent; this helps simplify computations significantly!

Find f'_y at the point M_0 = (2, 0, 4) for the function f(x, y, z) = e^(xyz) + (x+1)^3(y+1)^2 + sin(z-4).

Learn how to calculate the partial derivative f'_y at the point M_0 = (2, 0, 4) for the multivariable function f(x, y, z) = e^(xyz) + (x+1)^3(y+1)^2 + sin(z-4). Includes step-by-step solution using differentiation rules.

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