If y= ⁻¹ ( 1 1+x+x² )+ ⁻¹ ( 1 x²+3 x+3 )+ ⁻¹ ( 1 x²+5 x+7 )+ + upto n terms then d y d x at x=0 and n=1 is equal to

Step-by-step solution

We have, aligned &y= ⁻¹ ( 1 1+x(x+1) )+ & ⁻¹ ( 1 1+(x+2)(x+3) )+ & ⁻¹ ( 1 1+(x+1)(x+2) )+ &+ ⁻¹ ( 1 1+(x+n-1)(x+n) ) aligned When n=1 , then aligned y &= ⁻¹ ( 1 1+x(x+1) ) &= ⁻¹ ( (x+1)-x 1+x(x+1) ) y &= ⁻¹(x+1)- ⁻¹ x d y d x &= 1 1+(x+1)² - 1 1+x² . d y d x |_ x=0 &= 1 1+(0+1)² - 1 1+(0)² &= 1 1+1 - 1 1 = 1 2 -1=- 1 2 aligned