Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

De 2^de orde afgeleide $https://latex.codecogs.com/png.image?\dpi{110}D^2\, \frac{1}{x^{-1}-1}$ is gelijk aan	A. $\large\boldsymbol{\frac {1} {(1\,-\,x)^4} }$
	B. $\large\boldsymbol{\frac {2} {(1\,-\,x)^3} }$
	C. $\large\boldsymbol{\frac {2} {(x\,-\,1)^3} }$
	D. $\large\boldsymbol{\frac {-1} {(1\,-\,x)^4} }$
	E. $\large\boldsymbol{\frac {1\,+\,x} {{(1\,-\,x)^3}} }$
	F. $\large\boldsymbol{\frac {-2} {(1\,-\,x)^3} }$

[ 5-2422 - op net sinds 18.6.05-(E)-15.11.2024 ]

Translation in E N G L I S H

The second order derivative $https://latex.codecogs.com/png.image?\dpi{110}D^2\, \frac{1}{x^{-1}-1}$ is equal to	A. $\large\boldsymbol{\frac {1} {(1\,-\,x)^4} }$
	B. $\large\boldsymbol{\frac {2} {(1\,-\,x)^3} }$
	C. $\large\boldsymbol{\frac {2} {(x\,-\,1)^3} }$
	D. $\large\boldsymbol{\frac {-1} {(1\,-\,x)^4} }$
	E. $\large\boldsymbol{\frac {1\,+\,x} {{(1\,-\,x)^3}} }$
	F. $\large\boldsymbol{\frac {-2} {(1\,-\,x)^3} }$

Oplossing - Solution

$\begin{aligned} & D^2\, \frac{1}{x^{-1}-1}=D^2\, \frac{1}{\frac 1x-1} =D^2\, \frac{1}{\frac{1-x}{x}} =D^2\, \frac{x}{1-x} =D\,\frac{1.(1-x)-x.(-1)}{(1-x)^2}\\ &=D\,\frac{1-x+x}{(1-x)^2} =D\,\frac{1}{(1-x)^2}=-\frac{1}{(1-x)^4}.D(1-x)^2\\ & =-\frac{1}{(1-x)^4}.2(1-x)(-1)=\frac{2}{(1-x)^3} \end{aligned}$

De 2^de orde afgeleide $https://latex.codecogs.com/png.image?\dpi{110}D^2\, \frac{1}{x^{-1}-1}$ is gelijk aan	A. \(\large\boldsymbol{\frac {1} {(1\,-\,x)^4} }\)
	B. \(\large\boldsymbol{\frac {2} {(1\,-\,x)^3} }\)
	C. \(\large\boldsymbol{\frac {2} {(x\,-\,1)^3} }\)
	D. \(\large\boldsymbol{\frac {-1} {(1\,-\,x)^4} }\)
	E. \(\large\boldsymbol{\frac {1\,+\,x} {{(1\,-\,x)^3}} }\)
	F. \(\large\boldsymbol{\frac {-2} {(1\,-\,x)^3} }\)

The second order derivative $https://latex.codecogs.com/png.image?\dpi{110}D^2\, \frac{1}{x^{-1}-1}$ is equal to	A. \(\large\boldsymbol{\frac {1} {(1\,-\,x)^4} }\)
	B. \(\large\boldsymbol{\frac {2} {(1\,-\,x)^3} }\)
	C. \(\large\boldsymbol{\frac {2} {(x\,-\,1)^3} }\)
	D. \(\large\boldsymbol{\frac {-1} {(1\,-\,x)^4} }\)
	E. \(\large\boldsymbol{\frac {1\,+\,x} {{(1\,-\,x)^3}} }\)
	F. \(\large\boldsymbol{\frac {-2} {(1\,-\,x)^3} }\)