To solve this problem, let's calculate the partial derivatives of the function $f(x, y) = x^2 + 4x^3y$ with respect to $x$ and $y$, and then evaluate them at the point $(1, 4)$.

Step 1: Find $f_x(x, y)$

The partial derivative of $f$ with respect to $x$ is: $f_x(x, y) = \frac{\partial}{\partial x}(x^2 + 4x^3y)$ Differentiating term-by-term:

1. $\frac{\partial}{\partial x}(x^2) = 2x$
2. $\frac{\partial}{\partial x}(4x^3y) = 12x^2y$ (since $y$ is treated as a constant)

So, $f_x(x, y) = 2x + 12x^2y$

Step 2: Evaluate $f_x(1, 4)$

Substitute $x = 1$ and $y = 4$ into $f_x(x, y)$: $f_x(1, 4) = 2(1) + 12(1)^2(4) = 2 + 48 = 50$

Step 3: Find $f_y(x, y)$

The partial derivative of $f$ with respect to $y$ is: $f_y(x, y) = \frac{\partial}{\partial y}(x^2 + 4x^3y)$ Differentiating term-by-term:

1. $\frac{\partial}{\partial y}(x^2) = 0$ (since $x^2$ is independent of $y$)
2. $\frac{\partial}{\partial y}(4x^3y) = 4x^3$

So, $f_y(x, y) = 4x^3$

Step 4: Evaluate $f_y(1, 4)$

Substitute $x = 1$ into $f_y(x, y)$: $f_y(1, 4) = 4(1)^3 = 4$

Final Answers

$f_x(1, 4) = 50$ $f_y(1, 4) = 4$

Would you like a deeper explanation of any steps?

Here are some related questions:

1. What is the interpretation of partial derivatives in a real-world context?
2. How does the chain rule apply when differentiating functions of multiple variables?
3. What is the geometric meaning of $f_x$ and $f_y$ at a point on a surface?
4. How would the second partial derivatives of this function be calculated?
5. What happens to $f_x$ and $f_y$ if $y$ is treated as a function of $x$?

Tip: Remember that when finding partial derivatives, treat all other variables as constants with respect to the variable of differentiation.