Let $$ Z=x^{2} \sin \left(3 x+y^{3}\right) $$ (a) Evaluate $$ \frac{\partial z}{\partial x} $$ at $$ \left({ }^{2} / 3,0\right) $$ (b) Evaluate $$ Z_{y} $$ at $$ (1,1) $$

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Let's evaluate the partial derivatives of the function $$ Z = x^{2} \sin(3x + y^{3}) $$ as requested.

### Part (a): Evaluate $$ \frac{\partial Z}{\partial x} $$ at $$ \left(\frac{2}{3}, 0\right) $$

**Step 1: Compute the Partial Derivative with Respect to $$ x $$**

Given:
$$
Z = x^{2} \sin(3x + y^{3})
$$

Using the product rule for differentiation:
$$
\frac{\partial Z}{\partial x} = \frac{d}{dx}(x^{2}) \cdot \sin(3x + y^{3}) + x^{2} \cdot \frac{d}{dx}[\sin(3x + y^{3})]
$$

Compute each part:
$$
\frac{d}{dx}(x^{2}) = 2x
$$
$$
\frac{d}{dx}[\sin(3x + y^{3})] = \cos(3x + y^{3}) \cdot \frac{d}{dx}(3x + y^{3}) = 3 \cos(3x + y^{3})
$$

Thus:
$$
\frac{\partial Z}{\partial x} = 2x \sin(3x + y^{3}) + 3x^{2} \cos(3x + y^{3})
$$

**Step 2: Evaluate at $$ \left(\frac{2}{3}, 0\right) $$**

Plugging in $$ x = \frac{2}{3} $$ and $$ y = 0 $$:
$$
\frac{\partial Z}{\partial x}\Bigg|_{\left(\frac{2}{3},0\right)} = 2\left(\frac{2}{3}\right) \sin\left(3 \cdot \frac{2}{3} + 0^{3}\right) + 3\left(\frac{2}{3}\right)^{2} \cos\left(3 \cdot \frac{2}{3} + 0^{3}\right)
$$
$$
= \frac{4}{3} \sin(2) + \frac{12}{9} \cos(2)
$$
$$
= \frac{4}{3} \sin(2) + \frac{4}{3} \cos(2)
$$
$$
= \frac{4}{3} (\sin(2) + \cos(2))
$$

**Final Answer for Part (a):**
$$
\frac{\partial Z}{\partial x}\Bigg|_{\left(\frac{2}{3},0\right)} = \frac{4}{3} \left( \sin(2) + \cos(2) \right)
$$

---

### Part (b): Evaluate $$ Z_{y} $$ at $$ (1, 1) $$

**Step 1: Compute the Partial Derivative with Respect to $$ y $$**

Given:
$$
Z = x^{2} \sin(3x + y^{3})
$$

Differentiate with respect to $$ y $$:
$$
\frac{\partial Z}{\partial y} = x^{2} \cdot \frac{d}{dy}[\sin(3x + y^{3})] = x^{2} \cdot \cos(3x + y^{3}) \cdot \frac{d}{dy}(3x + y^{3})
$$
$$
= x^{2} \cdot \cos(3x + y^{3}) \cdot 3y^{2}
$$
$$
= 3x^{2} y^{2} \cos(3x + y^{3})
$$

**Step 2: Evaluate at $$ (1, 1) $$**

Plugging in $$ x = 1 $$ and $$ y = 1 $$:
$$
\frac{\partial Z}{\partial y}\Bigg|_{(1,1)} = 3(1)^{2} (1)^{2} \cos(3 \cdot 1 + 1^{3}) = 3 \cos(4)
$$

**Final Answer for Part (b):**
$$
Z_{y}\Big|_{(1,1)} = 3 \cos(4)
$$
**Part (a):** At $$ \left(\frac{2}{3}, 0\right) $$, the partial derivative $$ \frac{\partial Z}{\partial x} $$ is $$ \frac{4}{3} (\sin(2) + \cos(2)) $$. **Part (b):** At $$ (1, 1) $$, the partial derivative $$ Z_{y} $$ is $$ 3 \cos(4) $$.

Let
(a) Evaluate at
(b) Evaluate at

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