The problem requires calculating the first partial derivatives of the function $f(x, y) = 3x + 3y$ at the point $(1, 3)$.

Step-by-step Solution:

1. Partial derivative with respect to $x$: The partial derivative of $f(x, y)$ with respect to $x$ is found by treating $y$ as a constant: $\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(3x + 3y) = 3.$ At the point $(1, 3)$, the value of $\frac{\partial f}{\partial x}$ is still: $\frac{\partial f}{\partial x}(1, 3) = 3.$

2. Partial derivative with respect to $y$: The partial derivative of $f(x, y)$ with respect to $y$ is found by treating $x$ as a constant: $\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(3x + 3y) = 3.$ At the point $(1, 3)$, the value of $\frac{\partial f}{\partial y}$ is still: $\frac{\partial f}{\partial y}(1, 3) = 3.$

Final Answer:

$\frac{\partial f}{\partial x}(1, 3) = 3, \quad \frac{\partial f}{\partial y}(1, 3) = 3.$

Would you like further clarification or details? Here are some related questions to consider:

1. What does a partial derivative represent in terms of the function $f(x, y)$?
2. How do partial derivatives generalize to higher dimensions?
3. What would change if the function had nonlinear terms like $x^2$ or $xy$?
4. How are these results used in applications, such as optimization?
5. Can we visualize these derivatives geometrically?

Tip: When calculating partial derivatives, always focus on treating all other variables as constants, which simplifies the process.