The derivative of ⁻¹ ( 1+x^2 - 1 x ) with respect to ⁻¹ ( 2x 1+x^2 ) is...

Step-by-step solution

Let u = ⁻¹ ( 1+x^2 - 1 x ) and v = ⁻¹ ( 2x 1+x^2 ) . Substituting x = , we get: u = ⁻¹ ( 1+ ^2 - 1 ) u = ⁻¹ ( - 1 ) = ⁻¹ ( 1 - ) u = ⁻¹ ( 2 ^2( /2) 2 ( /2) ( /2) ) = ⁻¹ ( 2 ) = 2 = 1 2 ⁻¹x Now, for v : v = ⁻¹ ( 2 1+ ^2 ) = ⁻¹( 2 ) = 2 = 2 ⁻¹x Differentiating u and v with respect to x : du dx = 1 2(1+x^2) dv dx = 2 1+x^2 The derivative of u with respect to v is given by: du dv = du dx dv dx = 1 2(1+x^2) 2 1+x^2 = 1 4 Answer: 1 4