F R2R2 is defined by Fxyxyx2y Find dF12 using only the defi

F: R^2->R^2 is defined by F(x,y)=(xy,x-2y). Find dF(1,2) - using only the definition of differentiability.

Solution

Let f(x, y) be differentiable (or f has continuous partial derivatives) and if x = x(t), y = y(t) are differentiable functions on t, then the function w = f(x(t), y(t)) is differentiable at t and df dt = fx(x(t), y(t))x 0 (t) + fy(x(t), y(t))y 0 (t), i.e., df dt = f x dx dt + f y dy dt . Proof : By increment theorem we have f = fx(x0, y0)x + fy(x0, y0)y + 1x + 2y, 1, 2 0 as x, y 0 This implies that f t = fx x t + fy y t + 1 x t + 2 y t . Allow t 0, which implies that 1, 2 0 because x, y 0. Therefore, we get df dt = f x dx dt + f y