Find dy/dx and d2y/dx2 by implicit differentiation

[RyanKooper]

The question: Find dy/dx and d2y/dx2 by implicit differentiation when: x - y + 3xy = 2. I attached a pdf that shows how I tried to solve it. However the answer that I've come up with does not match the answer in the text book which is: dy/dx= (1+3y)/(1-3x)=-5/(1-3x)^2 and d2y/dx2=6y'/(1-3x)=-30/(1-3x)^3.

 

[Deleted member 4993]

The question: Find dy/dx and d2y/dx2 by implicit differentiation when: x - y + 3xy = 2. I attached a pdf that shows how I tried to solve it. However the answer that I've come up with does not match the answer in the text book which is: dy/dx= (1+3y)/(1-3x)=-5/(1-3x)^2 and d2y/dx2=6y'/(1-3x)=-30/(1-3x)^3.

Derive an expression for 'y'

x + y + 3xy = 2 ......\(\displaystyle \to \to \)...... y = (2-x)/(1 + 3x)

Now use this into the expressions that you got and simplify...

 

[Harry_the_cat]

Derive an expression for 'y'

x + y + 3xy = 2 ......\(\displaystyle \to \to \)...... y = (2-x)/(1 + 3x)

Now use this into the expressions that you got and simplify...

Yes, note the textbook's final answers involve only x. So you have done the differentiation implicitly (as the question dictates) but they seem to want the answers in terms of x only.

Note:
If the question didn't state that they wanted you to find the derivatives implicity, then you would do as Subho said right from the start and forget about doing it implicitly. You would get to the final answers (in terms of x only) much less painfully! Good way to check your answers too!

 

[JeffM]

I cannot get your pdf to render so I cannot see where or even whether you made an error.

Solving by implicit differentiation, I do get

[math]x - y + 3xy = 2 \implies y’ = \dfrac{1 + 3y}{1 - 3x} \implies y’’ = \dfrac{6(1 + 3y)}{(1 - 3x)^2} = \dfrac{6y’}{1 - 3x}.[/math]
Like our resident cat, I have no idea why, having asked for a derivation by implicit differentiation, they give answers in terms of only x.

 

[blamocur]

The question: Find dy/dx and d2y/dx2 by implicit differentiation when: x - y + 3xy = 2. I attached a pdf that shows how I tried to solve it. However the answer that I've come up with does not match the answer in the text book which is: dy/dx= (1+3y)/(1-3x)=-5/(1-3x)^2 and d2y/dx2=6y'/(1-3x)=-30/(1-3x)^3.

You did get (1+3y)/(1-3x) -- why do you say that the answer does not match?