To find the four second partial derivatives of the function f(x,y)=x + y - e^(x+y), you can begin by finding the first and second partial derivatives with respect to both x and y.
1. Partial Derivative of f with respect to x:
The partial derivative of f with respect to x (symbolized as f_x) involves taking the derivative of f(x,y) with regard to x, while treating y as a constant. This yields:
f_x = 1 - e^(x+y)
2. Second Partial Derivative of f with respect to x:
This involves deriving f_x with respect to x (symbolized as f_xx or ∂²f/∂x²). This gives us:
f_xx = -e^(x+y)
3. Partial Derivative of f with respect to y:
The process is similar to (1), but this time holding x as constant, resulting in:
f_y = 1 - e^(x+y)
4. Second Partial Derivative of f with respect to y:
Again, similar to (2), but this time deriving f_y with respect to y (symbolized as f_yy or ∂²f/∂y²). The end result is:
f_yy = -e^(x+y)
5. Cross Partial Derivatives:
These derivatives involve deriving with respect to both x and y. The order of the derivatives does not matter due to Schwarz's Theorem, which states that the cross derivatives of a sufficiently smooth function are equal, that is, f_xy = f_yx.
Computing f_xy, which involves taking the derivative of f_x with respect to y, gives:
f_xy = -e^(x+y)
Similarly, computing f_yx by taking the derivative of f_y with respect to x also gives:
f_yx = -e^(x+y)
This implies that the second cross partial derivatives of the function f are both equal to -e^(x+y).
In conclusion, the second partial derivatives of the function f(x,y) are as follows: f_xx = f_yy = f_xy = f_yx = -e^(x+y)