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 \( (1,1) \)

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):** \[ Z_{y}\Big|_{(1,1)} = 3 \cos(4) \]