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What is the derivative of `tan^(-1)((sqrtx-x)/(1+x^(3//2)))` at x = 1?A. `-(1)/(4)`B. `(1)/(2)`C. `(3)/(2)`D. 1

Answer» Correct Answer - A
Let `y=tan^(-1)((sqrtx-x)/(1+x^(3//2)))=tan^(-1).(sqrtx-x)/(1+sqrtx.x)`
`=tan^(-1)sqrtx-tan^(-1)x`
On differentiating w.r.t. we get
`(dy)/(dx)=(1)/(1+x).(1)/(2sqrtx)-(1)/(1+x^(2))`
Now, `((dy)/(dx))_(x=1)=(1)/(1+1).(1)/(2)-(1)/(1+1)=(1)/(4)-(1)/(2)=-(1)/(4).`


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