Question

Determine fₓ when
f(x, y)=2 x+y/x+2 y

Answer 1

The partial derivative of f(x, y) = (2x + y)/(x + 2y) with respect to x is obtained by applying the quotient rule, resulting in fₓ = (3y) / (x + 2y)².

Step-by-step explanation:

To determine fx when f(x, y) = (2x + y)/(x + 2y), we need to find the partial derivative of the function with respect to x. The partial derivative fx denotes the derivative of the function f with respect to x, holding other variables constant—in this case, holding y constant.

The partial derivative of f(x, y) is obtained by applying the quotient rule, which states that if we have a function in the form u/v, its derivative u'/v' = (vu' - uv') / v2. Applying the quotient rule:

fx = [(x + 2y)(2) - (2x + y)(1)] / (x + 2y)2

simplified: fx = (2x + 4y - 2x - y) / (x + 2y)2 = (3y) / (x + 2y)2.

Therefore, the partial derivative of f with respect to x is (3y) / (x + 2y)2.