Evaluate the first partial derivatives of the function at the given point.

f(x, y) = x^2 + xy + y^2 + 2x - y ;
at (-5, 4)
fx(-5, 4) = ?

fy(-5, 4) = ?

f(x, y) = x^2 + xy + y^2 + 2x - y ;

at (-5, 4)
fx(-5, 4) = 2x + y +2 = -10+ 4 + 2 = -4
fy(-5, 4) = x + 2y -1 = -5 + 8 -1 = 2

To find the first partial derivatives of the function at the given point, we need to differentiate the function with respect to each variable separately.

1. Finding fx(-5, 4):

To find fx, we differentiate the function f(x, y) with respect to x, treating y as a constant.

f(x, y) = x^2 + xy + y^2 + 2x - y

Differentiating with respect to x:

∂f/∂x = 2x + y + 2

Now, substituting x = -5 and y = 4:

fx(-5, 4) = 2(-5) + 4 + 2 = -10 + 4 + 2 = -4

Therefore, fx(-5, 4) = -4.

2. Finding fy(-5, 4):

To find fy, we differentiate the function f(x, y) with respect to y, treating x as a constant.

f(x, y) = x^2 + xy + y^2 + 2x - y

Differentiating with respect to y:

∂f/∂y = x + 2y - 1

Now, substituting x = -5 and y = 4:

fy(-5, 4) = (-5) + 2(4) - 1 = -5 + 8 - 1 = 2

Therefore, fy(-5, 4) = 2.

To evaluate the first partial derivatives of the function at the given point (-5, 4), we need to calculate the derivative with respect to each variable separately.

First, let's find the partial derivative with respect to x, denoted as fx(-5, 4).

To find fx, we differentiate the function f(x, y) = x^2 + xy + y^2 + 2x - y with respect to x, treating y as a constant. The derivatives of the various terms are as follows:

The derivative of x^2 with respect to x is 2x.
The derivative of xy with respect to x is y.
The derivative of y^2 with respect to x is 0 because y is treated as a constant.
The derivative of 2x with respect to x is 2.
The derivative of -y with respect to x is 0 because y is treated as a constant.

Now, we can sum up these derivatives to find fx:

fx(x, y) = 2x + y + 0 + 2 - 0 = 2x + y + 2

To evaluate fx(-5, 4), substitute x = -5 and y = 4 into the expression:

fx(-5, 4) = 2(-5) + 4 + 2 = -10 + 4 + 2 = -4.

Therefore, fx(-5, 4) = -4.

Now, let's find the partial derivative with respect to y, denoted as fy(-5, 4).

To find fy, we differentiate the function f(x, y) = x^2 + xy + y^2 + 2x - y with respect to y, treating x as a constant. The derivatives of the various terms are as follows:

The derivative of x^2 with respect to y is 0 because x is treated as a constant.
The derivative of xy with respect to y is x.
The derivative of y^2 with respect to y is 2y.
The derivative of 2x with respect to y is 0 because x is treated as a constant.
The derivative of -y with respect to y is -1.

Now, we can sum up these derivatives to find fy:

fy(x, y) = 0 + x + 2y + 0 - 1 = x + 2y - 1

To evaluate fy(-5, 4), substitute x = -5 and y = 4 into the expression:

fy(-5, 4) = (-5) + 2(4) - 1 = -5 + 8 - 1 = 2.

Therefore, fy(-5, 4) = 2.