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

 

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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

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