Differentiation - Product Rule

[quacky]

Homework Statement

In general I havn't had problems using the differentiation rules until I came on this question, I'm probably doing something stupid any help is handy. Plugged it into an online differentiation solver and it comes up with (x^2-1)/(x^2) which I am getting nowhere near to in my attempts.

Differentiate y = ((1/x)-1)*(1-x)2. The attempt at a solution

Using product rule

f(x)= x^-1 - 1
f'(x)= -x^-2
g(x)= 1-x
g'(x)= 1

f(x)*g'(x) = (x^-1 -1)(1) = x^-1 - 1
g(x)*f'(x) = (1-x)(-x^-2) = -x^-2 + x^-1

Adding them together = -x^-2 + 2x^-1 - 1

Simplifying slightly = -1 / (-x^2 + 2x)

And that's where I'm stuck, really not sure what I'm doing wrong here if its in the differentiation or messing up simplifying etc with algebra (been a long time since I've done math)

Help much appreciated
Thanks

 

[Hootenanny]

Welcome to Physics Forums.

f(x)= x^-1 - 1
f'(x)= -x^-2
g(x)= 1-x
g'(x)= 1

A little slip here, which unfortunately carries through: $g^\prime(x)=-1$.

 

[quacky]

Ah yep silly mistake. Just gave it another go, still not getting the answer I'm expecting though, can anyone confirm if the answer I think I should be getting (x^2 - 1)/(x^2) is correct or not as I might be chasing something that is wrong to begin with (although its much more likely it's me doing something wrong).

Attempt 2

f(x) = x^-1 - 1
f'(x) = -x^-2
g(x) = (1 - x)
g'(x) = -1

f(x)*g'(x) = -x^-1 + 1
g(x)*f'(x) = -x^-2 + x

Adding together

-x^-2 - x^-1 + x + 1

Simplifying

(x + 1)/(-x^2 - x)

 

[Hootenanny]

can anyone confirm if the answer I think I should be getting (x^2 - 1)/(x^2) is correct or not

Yes, that is correct.

Attempt 2

f(x) = x^-1 - 1
f'(x) = -x^-2
g(x) = (1 - x)
g'(x) = -1

f(x)*g'(x) = -x^-1 + 1
g(x)*f'(x) = -x^-2 + x

These final two lines are incorrect.

Do you have to use the product rule? If not, you may find it easier to multiply out the brackets before differentiating.