If y= ⁻¹ ( 1 1+x+x² )+ ⁻¹ ( 1 x²+2 x+3 )+ ⁻¹ ( 1 x²+5 x+7 )+ +n terms, then y^ (0) is

Step-by-step solution

Given, aligned y &= ⁻¹ ( 1 x²+x+1 )+ ⁻¹ ( 1 x²+2 x+3 ) &+ ⁻¹ ( 1 x²+5 x+7 )+ n terms ( aligned Now , ⁻¹ ( 1 x²+x+1 )= ( x+1-x 1+x(x+1) ) = ⁻¹( x +1)- ⁻¹ x From Eq. (i), we get aligned &y= [ ⁻¹(x+1)- ⁻¹ x ]+ [ ⁻¹(x+2) . & .- ⁻¹(x+1) ]+ + [ ⁻¹(x+n)- ⁻¹ n ] & y= ⁻¹(x+n)- ⁻¹(x) aligned On differentiating w.r.t. ' x ', we get aligned dy dx &= 1 1+( x + n )² - 1 1+ x ² dx )_ at x =0 &= 1 1+ n ² -1 &= 1- (1+ n ² ) 1+ n ² &= - n ² 1+ n ² aligned