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.