A) \[n!\]
B) \[(n-1)!\]
C) \[{{(-1)}^{n}}(n-1)!\]
D) \[{{(-1)}^{n}}n!\]
E) \[(n+1)!\]
Correct Answer: C
Solution :
\[y=\left( 1+\frac{1}{x} \right)\left( 1+\frac{2}{x} \right)\left( 1+\frac{3}{x} \right)....\left( 1+\frac{n}{x} \right)\] \[\frac{dy}{dx}=\left( -\frac{1}{{{x}^{2}}} \right)\left( 1+\frac{2}{x} \right)\left( 1+\frac{3}{x} \right)....\left( 1+\frac{n}{x} \right)+\] \[\left( 1+\frac{1}{x} \right)\left( -\frac{2}{{{x}^{2}}} \right)\left( 1+\frac{3}{x} \right).......\left( 1+\frac{n}{x} \right)+\] \[.....+\left( 1+\frac{1}{x} \right)\left( 1+\frac{2}{x} \right)\left( 1+\frac{3}{x} \right).....\left( -\frac{n}{{{x}^{2}}} \right)\] \[\therefore \] \[{{\left. \frac{dy}{dx} \right|}_{x=-1}}=(-1)(-1)(-2)(-3).....(1-n)\] \[={{(-1)}^{n}}(1)(2)(3)....(n-1)\] \[={{(-1)}^{n}}(n-1)!\]
You need to login to perform this action.
You will be redirected in
3 sec
