Final answer:
The question involves finding the first partial derivatives of the function f(x,y) = (2x-y)/(2x+y) at the point (2,3). The solution requires applying the quotient rule to find both partial derivatives with respect to x and y.
Step-by-step explanation:
The student is asking to find the first partial derivatives of the function f(x,y) = (2x−y)/(2x+y) at the point (x,y) = (2,3). To do this, we calculate the partial derivative with respect to x, denoted as fx, and the partial derivative with respect to y, denoted as fy. The process involves applying the quotient rule for derivatives to both partial derivatives.
For fx, the derivative of the numerator (2x - y) with respect to x is 2, and the derivative of the denominator (2x + y) with respect to x is 2. Applying the quotient rule, we get:
fx(x, y) = [(2)(2x + y) - (2x - y)(2)] / (2x + y)^2
Substituting the point (2,3) into this expression, we get fx(2,3).
For fy, the derivative of the numerator (2x - y) with respect to y is -1, and the derivative of the denominator (2x + y) with respect to y is 1. Applying the quotient rule, we get:
fy(x, y) = [(-1)(2x + y) - (2x - y)(1)] / (2x + y)^2
Substituting the point (2,3) into this expression, we get fy(2,3).