Differentiation anomoly-in need of setting straight

[DyslexicHobo]

Differentiation anomoly--in need of setting straight!

I was studying for the AP calculus exam when I came across this question, which seems to have multiple answers. I cannot seem to find what is wrong with it (nor can my calc teacher).

Find dy/dx if x + y = x * y

I went about this problem the "easy way out". Instead of implicitly differentiating, I figured I could do some algebra to simplify it to y = x / (x-1). I saw the answers, and only one of them did not have a "y" in the answer, so I picked that one. It was wrong.

When solved implicitly, a different answer is found: (y-1)/(1-x). This is STILL different than what our answer key has, which is (1-y)/(x-1).

Can someone please help resolve this? Thanks!

 

[matt grime]

What ever other problems you're having:

When solved implicitly, a different answer is found: (y-1)/(1-x). This is STILL different than what our answer key has, which is (1-y)/(x-1).

contains a mistake on your part. The two answers written there are the same.

 

[Noober]

If you differentiate $$y=\frac{x}{x-1}$$, you get $$\frac{dy}{dx}=\frac{(x-1)-x}{(x-1)^2}$$.

Divide by $$\frac{x-1}{x-1}: \frac{dy}{dx}=\frac{1-\frac{x}{x-1}}{x-1}$$

Substitute $$y=\frac{x}{x-1}$$: $$\frac{dy}{dx}=\frac{1-y}{x-1}$$

 

[Office_Shredder]

When solved implicitly, a different answer is found: (y-1)/(1-x). This is STILL different than what our answer key has, which is (1-y)/(x-1).

No... it's just negative on the top and bottom. Multiply your answer by 1 = -1/-1 and see what you get

 

[AlephZero]

I went about this problem the "easy way out". Instead of implicitly differentiating...

Isn't implicit differentiation "the easy way"?

1 + dy/dx = y + x dy/dx

1-y = (x-1)dy/dx

dy/dx = (1-y)/(x-1)

 

[DyslexicHobo]

Isn't implicit differentiation "the easy way"?

1 + dy/dx = y + x dy/dx

1-y = (x-1)dy/dx

dy/dx = (1-y)/(x-1)

What I meant is that, because I could see that it simplified to y = *stuff with no y*, I figured that dy/dx MUST not have a y in it. There was only one choice without one, so I chose that. It was the easy way out because I did not do any differentiating at all.

Also, I figured out my problem. Both answers were the same, but had a "y", and the other did not. When a substitution was made for "y", the answers were identical.

And yeah... I can't believe I didn't notice that the answer simply divided by a factor of (-1). Wow I feel dumb!


All here is well, thank you for the help.