If y= 1- ⁻¹ x 1+ ⁻¹ x , then ( dy d x ) at x=0 is

Step-by-step solution

y= 1- ⁻¹ x 1+ ⁻¹ x Differentiating w.r.t. x , we get aligned d y ~d x & = d ~d x ( 1- ⁻¹ x 1+ ⁻¹ x ) & = 1 2 1- ⁻¹ x 1+ ⁻¹ x ~d ~d x ( 1- ⁻¹ x 1+ ⁻¹ x ) & = 1+ ⁻¹ x 2 1- ⁻¹ x aligned aligned & (1+ ⁻¹ x ) d d x (1- ⁻¹ x )- (1- ⁻¹ x ) d d x (1+ ⁻¹ x ) (1+ ⁻¹ x )^2 & = 1+ ⁻¹ x 2 1- ⁻¹ x (1+ ⁻¹ x ) -1 1-x^2 - (1- ⁻¹ x ) 1 1-x^2 (1+ ⁻¹ x )^2 aligned aligned &= 1+ ⁻¹ x 2 1- ⁻¹ x 1 1-x^2 [-1- ⁻¹ x-1+ ⁻¹ x ] (1+ ⁻¹ x )^2 & ~d y ~d x = (-1) 1+ ⁻¹ x 1- ⁻¹ x ( 1-x^2 ) (1+ ⁻¹ x )^2 & ( ~d y ~d x )_ at x=0 = -1 1+ ⁻¹ 0 ( 1- ^1