What is the first and second derivatives of x/(x-1)^2?

We have the function f(x) = x/(x - 1)^2 and we have to
find the first and the second derivatives.

f(x) = x/(x -
1)^2

=> f(x) = x*(x -
1)^-2

Use the product rule to calculate the first
derivative

f'(x) = x*(-2)(x - 1)^(-3) + (x -
1)^(-2)

=> -2x*(x - 1)^(-3) + (x -
1)^(-2)

=> -2x/(x - 1)^3 + 1/(x -
1)^2

Again use the product rule to calculate the second
derivative

f'(x) = -2x*(x - 1)^(-3) + (x -
1)^(-2)

f''(x) = (-2)[(x - 1)^(-3) + (-3)*x*(x - 1)^(-4) +
(-2)*(x - 1)^(-3)

=> -2/(x - 1)^3 + 6x/(x - 1)^4 -
2/(x - 1)^3

=> -4/(x - 1)^3 + 6x/(x -
1)^4

The first derivative is -2x/(x - 1)^3 +
1/(x - 1)^2 and the second derivative is -4/(x - 1)^3 + 6x/(x -
1)^4