Answer to Question #275761 in Calculus for JAY

Question

1. D²((x+1)²/x-1)            ans, 8/(x-1)³
2. D(ln(x²-2x+1))            ans, 2/x-1

Answer 1

$1. \ D\big((x+1)^2/(x-1)\big)=\frac{(x-1)\cdot D(x+1)^2-(x+1)^2\cdot D(x-1)}{(x-1)^2}= \frac{(x-1)\cdot 2(x+1)-(x+1)^2\cdot 1 }{(x-1)^2 }= \frac{x^2-2x-3 }{(x-1)^2 }= \frac{(x^2-2x+1)-4}{(x-1)^2}= 1-\frac{4}{(x-1)^2}$

$D^2\big((x+1)^2/(x-1)\big)=D\big(1-4\cdot (x-1)^{-2})=D1-4D(x-1)^{-2}=-4\cdot (-2)\cdot (x-1)^{-3}=\frac{8}{(x-1)^3}$

$2.\ D\big((\ln (x^2-2x+1)\big)=D\big(\ln (x-1)^2\big)=D\big(2\ln |x-1|\big)=2D\ln |x-1|=\frac{2}{x-1}$

Answers: $1)\ \frac{8}{(x-1)^3};\ \ \ 2)\frac{2}{x-1}.$