Question

Derivadas

Determine fx(x,y) e fy(x,y) para:

f(x,y) = (x+y) / (x-y)

Answer 1

$f_x(x,y) = \frac{\partial f}{\partial x} \\\\f_x(x,y) = \frac{\partial f}{\partial y}$

lembrando que a regra do quociente é 
$\left( \frac{U}{V} \right)' = \frac{U'*V - U*V'}{V^2}$

temos
$f(x,y) = \frac{x+y}{x-y}$

calculando a derivada fx(x,y) 
$\bmatrix{U = x+y\\\\U' =1+0\\\\V=x-y\\\\V'=1-0\end$

colocando na regra do quociente
$f_x(x,y)= \frac{1*(x-y) -(x+y)*1}{(x-y)^2} \\\\f_x(x,y)= \frac{x-y -x-y}{(x-y)^2}= \frac{-2y}{(x-y)^2}$

fazendo o mesmo processo para derivar em fy(x,y)
$\bmatrix{U = x+y\\\\U' =0+1\\\\V=x-y\\\\V'=0-1\end$

$f_x(x,y)= \frac{1*(x-y)-(x+y)*(-1)}{(x-y)^2} \\\\f_x(x,y)= \frac{x-y+x+y}{(x-y)^2} = \frac{2x}{(x-y)^2}$

Answer 2

Resposta:

A.

f xy = f yx = 1

Explicação passo a passo:

Determinefxy e fyxonde f(x,y)= x+y+xy.​​​​​​