Find fx
 and fy
 for

f(x,y)=(1−x+y)ex+2y

To find the partial derivatives \( f_x \) and \( f_y \) of the function \( f(x, y) = (1 - x + y)e^x + 2y \), we'll differentiate the function with respect to \( x \) and \( y \) separately.

### 1. Finding \( f_x \) (the partial derivative with respect to \( x \)):

Given:
\[
f(x, y) = (1 - x + y)e^x + 2y
\]

Differentiate \( f(x, y) \) with respect to \( x \):

\[
f_x = \frac{\partial}{\partial x} \left[ (1 - x + y)e^x + 2y \right]
\]

Apply the product rule to the first term \((1 - x + y)e^x\):

\[
f_x = \frac{\partial}{\partial x}(1 - x + y) \cdot e^x + (1 - x + y) \cdot \frac{\partial}{\partial x}(e^x) + \frac{\partial}{\partial x}(2y)
\]

Now, compute each term:

- \(\frac{\partial}{\partial x}(1 - x + y) = -1\)
- \(\frac{\partial}{\partial x}(e^x) = e^x\)
- \(\frac{\partial}{\partial x}(2y) = 0\) (since \(2y\) is independent of \(x\))

Substitute these into the equation:

\[
f_x = -e^x + (1 - x + y)e^x + 0
\]

Simplify:

\[
f_x = (1 - x + y - 1)e^x
\]

\[
f_x = (-x + y)e^x
\]

### 2. Finding \( f_y \) (the partial derivative with respect to \( y \)):

Differentiate \( f(x, y) \) with respect to \( y \):

\[
f_y = \frac{\partial}{\partial y} \left[ (1 - x + y)e^x + 2y \right]
\]

Now, compute each term:

- \(\frac{\partial}{\partial y}(1 - x + y) = 1\)
- \(e^x\) is treated as a constant with respect to \(y\), so \(\frac{\partial}{\partial y}(e^x) = 0\)
- \(\frac{\partial}{\partial y}(2y) = 2\)

Substitute these into the equation:

\[
f_y = e^x + 2
\]

### Final Answers:

- \( f_x = (-x + y)e^x \)
- \( f_y = e^x + 2 \)

Would you like any further details or have any questions?

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Here are 8 questions you might find helpful:
1. How is the product rule applied in partial differentiation?
2. Can the function \( f(x, y) = (1 - x + y)e^x + 2y \) have any points of inflection?
3. What are the critical points of the function \( f(x, y) \)?
4. How do you classify the critical points using the second derivative test?
5. What happens if the function had another term involving both \( x \) and \( y \)?
6. How does the partial derivative \( f_x \) change if the function was \( f(x, y) = (1 - x + y)e^{xy} + 2y \)?
7. What if the function were \( f(x, y, z) \)? How would the partial derivatives change?
8. How do you find mixed partial derivatives \( f_{xy} \) or \( f_{yx} \)?

**Tip:** When taking partial derivatives, treat the variable you're differentiating with respect to as the only variable, treating all others as constants.

Learn how to find the partial derivatives fx and fy for the function f(x, y) = (1 - x + y)e^x + 2y. Detailed step-by-step solution with application of the product rule in partial differentiation. Suitable for advanced college-level students and professionals.

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