Considerthefunctionf(x,y)=x+x2y2+y:
(i) compute its partial derivatives with respect to x and y

Given the function \( f(x, y) = x + x^2y^2 + y \), let's compute the partial derivatives with respect to \( x \) and \( y \).

### (i) Partial derivative of \( f(x, y) \) with respect to \( x \):

To find \( \frac{\partial f}{\partial x} \), we treat \( y \) as a constant and differentiate the function with respect to \( x \):

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

Now, differentiate each term:

1. \( \frac{\partial}{\partial x} (x) = 1 \)
2. \( \frac{\partial}{\partial x} (x^2y^2) = 2xy^2 \) (since \( y^2 \) is treated as a constant)
3. \( \frac{\partial}{\partial x} (y) = 0 \) (since \( y \) is independent of \( x \))

So, the partial derivative of \( f \) with respect to \( x \) is:

\[
\frac{\partial f}{\partial x} = 1 + 2xy^2
\]

### (ii) Partial derivative of \( f(x, y) \) with respect to \( y \):

To find \( \frac{\partial f}{\partial y} \), we treat \( x \) as a constant and differentiate the function with respect to \( y \):

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

Now, differentiate each term:

1. \( \frac{\partial}{\partial y} (x) = 0 \) (since \( x \) is independent of \( y \))
2. \( \frac{\partial}{\partial y} (x^2y^2) = 2x^2y \) (since \( x^2 \) is treated as a constant)
3. \( \frac{\partial}{\partial y} (y) = 1 \)

So, the partial derivative of \( f \) with respect to \( y \) is:

\[
\frac{\partial f}{\partial y} = 2x^2y + 1
\]

### Summary:

- \( \frac{\partial f}{\partial x} = 1 + 2xy^2 \)
- \( \frac{\partial f}{\partial y} = 2x^2y + 1 \)

Would you like more details or have any further questions? Here are some related questions:

1. What is the second-order partial derivative of \( f(x, y) \) with respect to \( x \)?
2. What is the second-order partial derivative of \( f(x, y) \) with respect to \( y \)?
3. Can you compute the mixed partial derivative \( \frac{\partial^2 f}{\partial x \partial y} \)?
4. How does the total derivative of \( f(x, y) \) behave if \( x \) and \( y \) are functions of another variable \( t \)?
5. Can you find the critical points of \( f(x, y) \) using the first derivatives?

**Tip:** To check for critical points of a function of two variables, you need to set the first-order partial derivatives equal to zero and solve the system of equations.

Considerthefunctionf(x,y)=x+x2y2+y: (i) compute its partial derivatives with respect to x and y

Compute the partial derivatives of the multivariable function f(x, y) = x + x^2y^2 + y with respect to x and y. This solution covers key concepts in multivariable calculus, including functions of two variables and the basic rules of differentiation. Suitable for undergraduate-level calculus students.

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