If x^2 ⁻¹ y x -y^2 ⁻¹ x y =k , then ( d y d x )_ (1,1) =

Step-by-step solution

Given, x^2 ⁻¹ ( y x )-y^2 ⁻¹ ( x y )=K Differentiate with respect to x , we get. aligned & x^2 d ⁻¹ ( y x ) d x + ⁻¹ ( y x ) d (x^2 ) d x - y^2 d ⁻¹ ( x y ) d x & - ⁻¹ ( x y ) d (y^2 ) d x =0 & x^2 ( x^2 x^2+y^2 ) ( x d y d x -y x^2 )+ ⁻¹ ( y x ) 2 x & -y^2 ( y^2 x^2+y^2 ) ( y -x d y d x y^2 )- ⁻¹ ( x y ) 2 y=0 & ( 1 2 ) (1 y₁(1,1)-1 )+ ⁻¹ ( 1 1 ) 2 & - ( 1 2 ) (1-y₁(1,1) )- ⁻¹ ( 1 1 ) 2=0 aligned aligned & (y₁ (1₁, 1 )-1 )=0 & y₁(1,1)=1 d y d x (1,1)=1 aligned