Implicit differentiation of xy = x - y

[hank]

Not sure if I'm getting the right answer here, so I could use some confirmation.

I am supposed to implicitly differentiate the following:
xy = x - y

Ok, so here we go..

d/dx[xy] = d/dx[x - y]

x*dy/dx + y = 1 - dy/dx //Applying product rule to the xy
x * dy/dx + dy/dx = 1 - y //I'm not sure I'm handling this y correctly.
dy/dx (x + 1) = 1 -y
dy/dx = (1 - y) / (x + 1) //Final answer

Is this correct?

 

[stapel]

Looks good to me.

You can check (when possible) by solving the original equation for "y=", differentiating, and comparing the results:

. . . . .xy = x - y
. . . . .xy + y = x

. . . . .y(x + 1) = x
. . . . .y = x / (x + 1)

. . . . .dy/dx = [1(x + 1) - x(1)] / (x + 1)²
. . . . . . . . . .= [x + 1 - x] / (x + 1)²
. . . . . . . . . .= 1 / (x + 1)²

Turning to your result, and plugging in with our "y=" from above, we get:

. . . . .dy/dx = (1 - y) / (x + 1)
. . . . . . . . . .= (1 - (x / (x + 1)) / (x + 1)
. . . . . . . . . .= [(1(x + 1) - x) / (x + 1)] / (x + 1)
. . . . . . . . . .= [(x + 1 - x) / (x + 1)] / (x + 1)
. . . . . . . . . .= [1 / (x + 1)] / (x + 1)
. . . . . . . . . .= 1 / (x + 1)²

So the derivative simplifies to be the same thing, either way you go.

Eliz.