We are given the function $g(x, y) = \sqrt{10 - xy}$. Our task is to find the partial derivatives of $g$ with respect to $x$ and $y$, and then evaluate them at the point $(x, y) = (1.5, 3.5)$.

Step 1: Compute the partial derivative of $g(x, y)$ with respect to $x$

The function is $g(x, y) = \sqrt{10 - xy}$, so we use the chain rule to differentiate with respect to $x$.

$g_x = \frac{\partial}{\partial x} \left( \sqrt{10 - xy} \right) = \frac{1}{2 \sqrt{10 - xy}} \cdot (-y)$

So,

$g_x = \frac{-y}{2 \sqrt{10 - xy}}$

Step 2: Compute the partial derivative of $g(x, y)$ with respect to $y$

Now, we differentiate $g(x, y)$ with respect to $y$, again using the chain rule:

$g_y = \frac{\partial}{\partial y} \left( \sqrt{10 - xy} \right) = \frac{1}{2 \sqrt{10 - xy}} \cdot (-x)$

So,

$g_y = \frac{-x}{2 \sqrt{10 - xy}}$

Step 3: Evaluate $g_x$ and $g_y$ at $(x, y) = (1.5, 3.5)$

Now, we substitute $x = 1.5$ and $y = 3.5$ into the expressions for $g_x$ and $g_y$.

First, calculate $10 - xy$ at $(x, y) = (1.5, 3.5)$:

$10 - 1.5 \times 3.5 = 10 - 5.25 = 4.75$

Thus,

$\sqrt{10 - 1.5 \times 3.5} = \sqrt{4.75} \approx 2.179$

Now calculate $g_x$ and $g_y$:

$g_x = \frac{-3.5}{2 \times 2.179} = \frac{-3.5}{4.358} \approx -0.803$

$g_y = \frac{-1.5}{2 \times 2.179} = \frac{-1.5}{4.358} \approx -0.344$

Final Results:

• $g_x(1.5, 3.5) \approx -0.803$
• $g_y(1.5, 3.5) \approx -0.344$

Would you like further details or clarifications on any step?

Here are some related questions:

1. How would the partial derivatives change if the function were $g(x, y) = \sqrt{x^2 + y^2}$?
2. Can you explain how to apply the chain rule in partial derivatives?
3. What is the geometric interpretation of partial derivatives in multivariable calculus?
4. How do you compute second-order partial derivatives like $g_{xx}$ or $g_{xy}$?
5. How do you evaluate partial derivatives at boundary points or singularities?

Tip: When differentiating a function of multiple variables, always remember to treat the other variables as constants when taking partial derivatives.