Question

A expressão da derivada y' para a função y = f(x) dada implicitamente pela expressão x³ = x+y / x-y

Answer 1

Derivar implicitamente eh igualzinho derivar normal, mas toda vez que vc deriva o y vc multiplica o resultado por y':

$x^3=\dfrac{x+y}{x-y}\\\\\\3x^2=\dfrac{(x+y)'(x-y)-(x-y)'(x+y)}{(x-y)^2}\\\\\\3x^2=\dfrac{(1+y')(x-y)-(1-y')(x+y)}{(x-y)^2}\\\\\\3x^2=\dfrac{x-y+xy'-yy'-(x+y-xy'-yy')}{(x-y)^2}\\\\\\3x^2(x-y)^2=x-y+xy'-yy'-x-y+xy'+yy'\\\\\\3x^2(x-y)^2=2xy'-2y\\\\2xy'=3x^2(x-y)^2+2y\\\\\\y'=\dfrac{3x^2(x-y)^2+2y}{2x}$