For the function of two variables
f(x, y) = 4x
3 + 2x
2
y
3 + 3x
2
y + 2y
5
the first order partial derivatives with respect to x and y are:
(a) fx = 12x + 4xy3 + 6xy + 10y
4
, fy = 12x
2 + 6x
2
y
2 + 3x
2 + 10y
4
(b) fx = 12x + 4xy3 + 6xy, fy = 6x
2
y
2 + 3x
2 + 10y
4
(c) fx = 12x
2 + 4xy3 + 6xy, fy = 6x
2
y
2 + 3x
2 + 10y
4
(d) fx = 12x
2 + 4xy3 + 6xy, fy = 6x
2
y
2 + 3x
2
y + 10y
4
(e) none of these


Let's first rewrite the function \( f(x, y) \) in a clear format:

\[
f(x, y) = 4x^3 + 2x^2y^3 + 3x^2y + 2y^5
\]

Now, let's calculate the first-order partial derivatives with respect to \( x \) and \( y \).

### Partial Derivative with Respect to \( x \):

\[
f_x(x, y) = \frac{\partial}{\partial x} \left( 4x^3 + 2x^2y^3 + 3x^2y + 2y^5 \right)
\]

1. For the term \( 4x^3 \), the derivative with respect to \( x \) is:
   \[
   \frac{\partial}{\partial x} (4x^3) = 12x^2
   \]
   
2. For the term \( 2x^2y^3 \), treating \( y^3 \) as a constant:
   \[
   \frac{\partial}{\partial x} (2x^2y^3) = 4xy^3
   \]

3. For the term \( 3x^2y \), treating \( y \) as a constant:
   \[
   \frac{\partial}{\partial x} (3x^2y) = 6xy
   \]

4. For the term \( 2y^5 \), which does not contain \( x \):
   \[
   \frac{\partial}{\partial x} (2y^5) = 0
   \]

Adding these up:

\[
f_x(x, y) = 12x^2 + 4xy^3 + 6xy
\]

### Partial Derivative with Respect to \( y \):

\[
f_y(x, y) = \frac{\partial}{\partial y} \left( 4x^3 + 2x^2y^3 + 3x^2y + 2y^5 \right)
\]

1. For the term \( 4x^3 \), which does not contain \( y \):
   \[
   \frac{\partial}{\partial y} (4x^3) = 0
   \]

2. For the term \( 2x^2y^3 \), treating \( x^2 \) as a constant:
   \[
   \frac{\partial}{\partial y} (2x^2y^3) = 6x^2y^2
   \]

3. For the term \( 3x^2y \), treating \( x^2 \) as a constant:
   \[
   \frac{\partial}{\partial y} (3x^2y) = 3x^2
   \]

4. For the term \( 2y^5 \):
   \[
   \frac{\partial}{\partial y} (2y^5) = 10y^4
   \]

Adding these up:

\[
f_y(x, y) = 6x^2y^2 + 3x^2 + 10y^4
\]

### Conclusion:
Comparing these derivatives with the options:

- \( f_x(x, y) = 12x^2 + 4xy^3 + 6xy \)
- \( f_y(x, y) = 6x^2y^2 + 3x^2 + 10y^4 \)

The correct answer is **(b)**. 

Do you want more details or have any questions? 

Here are some related questions to consider:
1. How do you find higher-order partial derivatives?
2. What is the geometric interpretation of partial derivatives?
3. How can the chain rule be applied in functions of multiple variables?
4. What are mixed partial derivatives, and how do they relate to each other?
5. How does the gradient vector relate to the function’s partial derivatives?
6. Can you solve a similar problem with different functions?
7. What does it mean for a function to be differentiable with respect to multiple variables?
8. How do critical points relate to partial derivatives?

**Tip:** Always simplify the function as much as possible before differentiating; it can often reveal easier paths to finding derivatives.

Learn how to calculate first-order partial derivatives of a multivariable function f(x, y) = 4x^3 + 2x^2y^3 + 3x^2y + 2y^5. Detailed step-by-step solution with formulas and examples provided. Suitable for advanced college-level students and learners of multivariable calculus.

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