Final answer:
To find fx(2, -3), we differentiate the function F(x, y) = √(2x^2 + 2y^2) with respect to x and evaluate at the point (2, -3), resulting in fx(2, -3) = 4/√(26).
Step-by-step explanation:
The question involves calculating the partial derivative of the function F(x,y) with respect to x, evaluated at the point (2, -3).
To find fx(2, -3), we need to differentiate the function F(x,y) partially with respect to x while treating y as a constant and then plug in the values for x and y.
The given function is F(x,y) = √(2x^2 + 2y^2). The partial derivative with respect to x is:
fx(x,y) = ∂/∂x (√(2x^2 + 2y^2))
= 1/2(2x^2 + 2y^2)^{-1/2}*4x
= 2x/√(2x^2 + 2y^2)
Substituting x = 2 and y = -3 into the derivative, we get:
fx(2,-3) = 2*2/√(2*2^2 + 2*(-3)^2)
= 4/√(8 + 18)
= 4/√(26)
= 4/√(2*13)
= 4/√(26)