Find both first partial derivatives. $$ z=15 x \sqrt{y} $$

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UpStudy AI · Accepted Answer

We are given the function

$$
z = 15x \sqrt{y} = 15x y^{\frac{1}{2}}.
$$

### Partial derivative with respect to $$ x $$

To find $$\frac{\partial z}{\partial x}$$, we treat $$ y $$ as a constant. Differentiating with respect to $$ x $$:

$$
\frac{\partial z}{\partial x} = 15 \sqrt{y}.
$$

### Partial derivative with respect to $$ y $$

To find $$\frac{\partial z}{\partial y}$$, we treat $$ x $$ as a constant. The function in terms of $$ y $$ is

$$
z = 15x y^{\frac{1}{2}}.
$$

Differentiate with respect to $$ y $$ using the power rule:

$$
\frac{d}{dy} y^{\frac{1}{2}} = \frac{1}{2} y^{-\frac{1}{2}}.
$$

Thus, the partial derivative is:

$$
\frac{\partial z}{\partial y} = 15x \cdot \frac{1}{2} y^{-\frac{1}{2}} = \frac{15x}{2} \cdot \frac{1}{\sqrt{y}} = \frac{15x}{2\sqrt{y}}.
$$

### Final Answers

$$
\frac{\partial z}{\partial x} = 15\sqrt{y}
$$

$$
\frac{\partial z}{\partial y} = \frac{15x}{2\sqrt{y}}
$$
The partial derivatives are: $$ \frac{\partial z}{\partial x} = 15\sqrt{y} $$ $$ \frac{\partial z}{\partial y} = \frac{15x}{2\sqrt{y}} $$

Find both first partial derivatives. \[ z=15 x \sqrt{y} \]

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