To solve this, we need to compute the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for the given function $f(x, y) = x^2 + 3xy + y - 1$, and evaluate them at the point $(4, -5)$.

Step 1: Compute $\frac{\partial f}{\partial x}$

The partial derivative of $f(x, y)$ with respect to $x$ is: $\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( x^2 + 3xy + y - 1 \right)$ Differentiating term by term:

• $\frac{\partial}{\partial x}(x^2) = 2x$,
• $\frac{\partial}{\partial x}(3xy) = 3y$ (since $y$ is treated as a constant),
• $\frac{\partial}{\partial x}(y) = 0$,
• $\frac{\partial}{\partial x}(-1) = 0$.

Thus: $\frac{\partial f}{\partial x} = 2x + 3y$

Step 2: Compute $\frac{\partial f}{\partial y}$

The partial derivative of $f(x, y)$ with respect to $y$ is: $\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left( x^2 + 3xy + y - 1 \right)$ Differentiating term by term:

• $\frac{\partial}{\partial y}(x^2) = 0$ (since $x^2$ is constant with respect to $y$),
• $\frac{\partial}{\partial y}(3xy) = 3x$,
• $\frac{\partial}{\partial y}(y) = 1$,
• $\frac{\partial}{\partial y}(-1) = 0$.

Thus: $\frac{\partial f}{\partial y} = 3x + 1$

Step 3: Evaluate at $(x, y) = (4, -5)$

1. For $\frac{\partial f}{\partial x}$: $\frac{\partial f}{\partial x} = 2x + 3y = 2(4) + 3(-5) = 8 - 15 = -7$

2. For $\frac{\partial f}{\partial y}$: $\frac{\partial f}{\partial y} = 3x + 1 = 3(4) + 1 = 12 + 1 = 13$

Final Answer:

$\frac{\partial f}{\partial x} = -7, \quad \frac{\partial f}{\partial y} = 13$

Do you want further clarifications or examples?
Here are five follow-up questions to extend your learning:

1. How do partial derivatives relate to the concept of gradient vectors?
2. What does the value of $\frac{\partial f}{\partial x}$ represent geometrically at a point?
3. How would the second-order partial derivatives for $f(x, y)$ be computed?
4. Can you explain why $\frac{\partial}{\partial x}(3xy)$ includes the $y$ term?
5. How would the result change if the function were $f(x, y) = x^2 + 3xy^2 + y - 1$?

Tip: Partial derivatives are crucial in optimization problems and help analyze how a function changes with respect to each variable independently.