The derivative of ⁻¹ [ x 1+ x ] with respect to ⁻¹ [ x 1+ x ] is

Step-by-step solution

Let u= ⁻¹ ( x 1+ x ), v= ⁻¹ ( x 1+ x ) aligned u &= ⁻¹ ( 2 x / 2 x / 2 1+2 ² x / 2-1 ) &= ⁻¹( x / 2)=x / 2 aligned v= ⁻¹ ( x ² x 2 + ² x 2 +2 x 2 x 2 )= ⁻¹ ( ( ² x 2 - ² x 2 ) ( x 2 + x 2 )² )= ⁻¹ ( x 2 - x 2 x 2 + x 2 )= ⁻¹ ( 1- x 2 1+ x 2 ) aligned u &= ⁻¹ ( 2 x / 2 x / 2 1+2 ² x / 2-1 ) &= ⁻¹( x / 2)=x / 2 aligned v= ⁻¹ ( x ² x 2 + ² x 2 +2 x 2 x 2 )= ⁻¹ ( ( ² x 2 - ² x 2 ) ( x 2 + x 2 )² )= ⁻¹ ( x 2 - x 2 x 2 + x 2 )= ⁻¹ ( 1- x 2 1+ x 2 )= [ ⁻¹ ( ⁻¹ ( 4 - 2 ) ]= ( 4 - x 2 ) . Now, d u d x = 1 2 and d v d x = 1