The correct option is B −n21+n2
y=tan−1{11+x(1+x)}+tan−1{11+(x+1)(x+2)}+tan−1{11+(x+2)(x+3)}+......+tan−1{11+(x+n−1)(x+n)}=∑nr=1tan−1{11+(x+r−1)(x+r)}=∑nr=1tan−1{(x+r)−(x+r−1)1+(x+r−1)(x+r)}=∑nr=1{tan−1(x+r)−tan−1(x+r−1)}=tan−1(x+n)−tan−1xy′=11+(x+n)2−11+x2⇒y′(0)=11+n2−1=−n21+n2