Find fx
 and fy
 for

f(x,y)=3x2+2x2y2+3xy3+2y4

To find the partial derivatives \( f_x \) and \( f_y \) for the function

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

we differentiate with respect to \( x \) and \( y \) separately.

### 1. Partial derivative with respect to \( x \) (\( f_x \)):
\[
f_x(x, y) = \frac{\partial}{\partial x} \left( 3x^2 + 2x^2y^2 + 3xy^3 + 2y^4 \right)
\]

- For the term \( 3x^2 \), the derivative with respect to \( x \) is \( 6x \).
- For the term \( 2x^2y^2 \), \( y^2 \) is treated as a constant, so the derivative is \( 4xy^2 \).
- For the term \( 3xy^3 \), using the product rule, the derivative with respect to \( x \) is \( 3y^3 \).
- For the term \( 2y^4 \), since there is no \( x \) involved, the derivative is \( 0 \).

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

### 2. Partial derivative with respect to \( y \) (\( f_y \)):
\[
f_y(x, y) = \frac{\partial}{\partial y} \left( 3x^2 + 2x^2y^2 + 3xy^3 + 2y^4 \right)
\]

- For the term \( 3x^2 \), since there is no \( y \) involved, the derivative is \( 0 \).
- For the term \( 2x^2y^2 \), \( x^2 \) is treated as a constant, so the derivative is \( 4x^2y \).
- For the term \( 3xy^3 \), using the product rule, the derivative with respect to \( y \) is \( 9xy^2 \).
- For the term \( 2y^4 \), the derivative with respect to \( y \) is \( 8y^3 \).

So,
\[
f_y(x, y) = 4x^2y + 9xy^2 + 8y^3
\]

### Final results:

\[
f_x(x, y) = 6x + 4xy^2 + 3y^3
\]
\[
f_y(x, y) = 4x^2y + 9xy^2 + 8y^3
\]

Would you like more details on these calculations, or do you have any other questions?

### 8 Related Questions:
1. How do you find higher-order partial derivatives?
2. What is the significance of partial derivatives in multivariable functions?
3. Can you apply the product rule in partial differentiation?
4. How does the chain rule work in partial differentiation?
5. What are critical points in multivariable calculus?
6. How do you interpret \( f_x \) and \( f_y \) geometrically?
7. How do you find the mixed partial derivatives \( f_{xy} \) and \( f_{yx} \)?
8. How do you compute the gradient of a function?

### Tip:
Always check if a function is differentiable before applying differentiation rules, especially in the context of partial derivatives.

Learn how to find partial derivatives fx and fy for the multivariable function f(x, y) = 3x^2 + 2x^2y^2 + 3xy^3 + 2y^4. This problem involves applying partial differentiation techniques, including the product rule, in multivariable calculus. Suitable for undergraduate students studying calculus.

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