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Let's start solving the problems one by one from the assignment. Here, we'll tackle question 1 first.

### Question 1
**Problem:** Find \( \frac{\partial}{\partial z} \) at (1,2,3) for \( f(x,y,z) = x^2 y z^2 \).

**Solution:**
To find \( \frac{\partial f}{\partial z} \), we need to differentiate \( f(x, y, z) = x^2 y z^2 \) with respect to \( z \) while treating \( x \) and \( y \) as constants.

The partial derivative of \( f \) with respect to \( z \) is:
\[ \frac{\partial}{\partial z}(x^2 y z^2) = x^2 y \cdot 2z \]
\[ \frac{\partial f}{\partial z} = 2x^2 y z \]

Now, substituting \( x = 1 \), \( y = 2 \), and \( z = 3 \) into the derivative:
\[ \frac{\partial f}{\partial z} \bigg|_{(1,2,3)} = 2 \cdot 1^2 \cdot 2 \cdot 3 = 12 \]

So, \( \frac{\partial f}{\partial z} \) at the point (1,2,3) is **12**.

Let me know if you'd like to proceed to the next question or if you have any questions about this solution! Here are a few related questions you might consider:
1. How would you find the second derivative with respect to \( z \) for the function \( f(x, y, z) \)?
2. Could you solve for \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) at the same point?
3. What is the importance of evaluating derivatives at specific points?
4. How would this solution change if \( f \) were a function of four variables instead of three?
5. What applications might this type of problem have in fields like engineering or physics?

**Tip:** When solving for partial derivatives, always remember to treat other variables as constants except the one with respect to which you are differentiating.

Solve all questions in the assignment on partial differentiation.

Solve advanced calculus problems involving partial differentiation, Euler's Theorem, and trigonometric identities. Learn step-by-step solutions to find partial derivatives, prove Euler’s Theorem for homogeneous functions, and apply modified theorems in specific cases. Suitable for college students.

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