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To solve the problem, we need to calculate the following partial derivatives for each function \( y \) in parts (a), (b), and (c):

1. First-order partial derivatives:
   - \( \frac{\partial y}{\partial x} \)
   - \( \frac{\partial y}{\partial z} \)

2. Second-order partial derivatives:
   - \( \frac{\partial^2 y}{\partial x^2} \)
   - \( \frac{\partial^2 y}{\partial z^2} \)
   - Mixed partial derivative \( \frac{\partial^2 y}{\partial x \partial z} \)

Let's go through each part in turn.

### Part (a) \( y = 6 + 3x + 16z + 4x^2 + 2z^2 \)
1. **First-order partial derivatives**:
   - \( \frac{\partial y}{\partial x} = 3 + 8x \)
   - \( \frac{\partial y}{\partial z} = 16 + 4z \)

2. **Second-order partial derivatives**:
   - \( \frac{\partial^2 y}{\partial x^2} = 8 \)
   - \( \frac{\partial^2 y}{\partial z^2} = 4 \)
   - \( \frac{\partial^2 y}{\partial x \partial z} = 0 \)

### Part (b) \( y = 14x^3 z^2 \)
1. **First-order partial derivatives**:
   - \( \frac{\partial y}{\partial x} = 42x^2 z^2 \)
   - \( \frac{\partial y}{\partial z} = 28x^3 z \)

2. **Second-order partial derivatives**:
   - \( \frac{\partial^2 y}{\partial x^2} = 84x z^2 \)
   - \( \frac{\partial^2 y}{\partial z^2} = 28x^3 \)
   - \( \frac{\partial^2 y}{\partial x \partial z} = 84x^2 z \)

### Part (c) \( y = 9 + 4xz - 3x^{-2}z^3 \)
1. **First-order partial derivatives**:
   - \( \frac{\partial y}{\partial x} = 4z + 6x^{-3}z^3 \)
   - \( \frac{\partial y}{\partial z} = 4x - 9x^{-2}z^2 \)

2. **Second-order partial derivatives**:
   - \( \frac{\partial^2 y}{\partial x^2} = -18x^{-4}z^3 \)
   - \( \frac{\partial^2 y}{\partial z^2} = -18x^{-2}z \)
   - \( \frac{\partial^2 y}{\partial x \partial z} = 4 - 9x^{-3}z^2 \)

Would you like further details on any specific part or derivative? 

Here are some related questions that might deepen your understanding:

1. How would the results change if we had different exponents on \( x \) and \( z \) in each function?
2. What rules apply when calculating mixed partial derivatives for functions of multiple variables?
3. Can the second mixed derivative \( \frac{\partial^2 y}{\partial x \partial z} \) be different from \( \frac{\partial^2 y}{\partial z \partial x} \)?
4. How does the chain rule apply to functions of more than one variable?
5. How would the derivatives look if an additional variable, say \( w \), was introduced?

**Tip:** When calculating partial derivatives, treat all variables except the one you're differentiating with respect to as constants.

Tentukan ∂y/∂x, ∂y/∂z, ∂²y/∂x², ∂²y/∂z², dan ∂²y/∂x∂z dari fungsi majemuk berikut untuk y sebagai fungsi dari x dan z.

Learn how to calculate first-order and second-order partial derivatives for multivariable functions. This example provides a step-by-step solution for functions of x and z, covering core concepts in partial differentiation and the use of mixed derivatives.

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