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To solve this problem, we need to find the partial derivatives \( f_x(x,y) \) and \( f_y(x,y) \) of the given function \( f(x,y) = e^{x+y} + 7 \), and then evaluate them at specific points.

1. **Finding \( f_x(x,y) \)**:

The partial derivative with respect to \( x \), denoted \( f_x(x,y) \), is found by differentiating \( f(x,y) \) with respect to \( x \) while treating \( y \) as a constant.

\[ f(x,y) = e^{x+y} + 7 \]

\[ f_x(x,y) = \frac{\partial}{\partial x} (e^{x+y} + 7) \]

The constant term \( 7 \) disappears when differentiating.

\[ f_x(x,y) = \frac{\partial}{\partial x} (e^{x+y}) \]

Since the exponent \( x + y \) is treated as a whole and \( y \) is a constant, we get:

\[ f_x(x,y) = e^{x+y} \]

2. **Finding \( f_y(x,y) \)**:

The partial derivative with respect to \( y \), denoted \( f_y(x,y) \), is found by differentiating \( f(x,y) \) with respect to \( y \) while treating \( x \) as a constant.

\[ f_y(x,y) = \frac{\partial}{\partial y} (e^{x+y} + 7) \]

Again, the constant term \( 7 \) disappears when differentiating.

\[ f_y(x,y) = \frac{\partial}{\partial y} (e^{x+y}) \]

Since the exponent \( x + y \) is treated as a whole and \( x \) is a constant, we get:

\[ f_y(x,y) = e^{x+y} \]

3. **Evaluating \( f_x(2, -1) \)**:

Using the partial derivative \( f_x(x,y) = e^{x+y} \), we substitute \( x = 2 \) and \( y = -1 \):

\[ f_x(2, -1) = e^{2 + (-1)} \]
\[ f_x(2, -1) = e^{1} \]
\[ f_x(2, -1) = e \]

4. **Evaluating \( f_y(-4, 3) \)**:

Using the partial derivative \( f_y(x,y) = e^{x+y} \), we substitute \( x = -4 \) and \( y = 3 \):

\[ f_y(-4, 3) = e^{-4 + 3} \]
\[ f_y(-4, 3) = e^{-1} \]
\[ f_y(-4, 3) = \frac{1}{e} \]

**Summary of Results:**

- \( f_x(x,y) = e^{x+y} \)
- \( f_y(x,y) = e^{x+y} \)
- \( f_x(2, -1) = e \)
- \( f_y(-4, 3) = \frac{1}{e} \)

Would you like any more details or have any questions?

### Related Questions:

1. What is the general method for finding partial derivatives of exponential functions?
2. How do we differentiate exponential functions with multiple variables?
3. Why do constant terms disappear when differentiating?
4. How would the result change if the function was \( f(x,y) = e^{x^2 + y^2} + 7 \)?
5. Can you explain the importance of evaluating partial derivatives at specific points?
6. What is the significance of the base \( e \) in calculus?
7. How do partial derivatives help in understanding the behavior of multivariable functions?
8. How would you find the second-order partial derivatives of \( f(x,y) \)?

### Tip:

When dealing with exponential functions in partial derivatives, always apply the chain rule appropriately, treating each variable independently while holding the others constant. This ensures accurate differentiation and helps in understanding the function's behavior.

Explore how to find partial derivatives of exponential functions like \( f(x,y) = e^{x+y} + 7 \). This detailed guide covers the differentiation process, including the chain rule application. Learn to evaluate these derivatives at specific points like (2, -1) and (-4, 3).

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