I have this expression $$\frac{1}{(1 - x) ^ 2}$$ I need the derivative of this expression. So I calculated it, no big deal. However something has crossed my mind. Mathematically $(1 - x) ^ 2 = (x - 1) ^ 2$ isn't it ? So why is the derivatives of $1 / (1 - x) ^ 2$ and $1 / (x - 1) ^ 2$ are different ? Given that we take the derivative of the function inside, the minus sign will appear in the first expression but not in the second. I would be grateful if somebody could explain the behavior to me.
I calculated the results of these derivatives and confirmed from the website http://www.derivative-calculator.net/
The derivative of $1 / (x - 1)^2$ is $-2/(x-1)^3$
And the derivative of $1 / (1 - x)^2$ is $2/(x-1)^3$
Aren't they different ? Am I missing something ?
It isn't different.
There's a minus sign introduced by the derivative of (1-x), but there's also an odd power of (1-x), which is the second minus sign when compared to (x-1)