[x^(y)+y^(x)=1],[(dy)/(dz)=?]

If y = (lot _(e) x)/(x) and z = log _(e) x, then (d ^(2) y)/(dz^(2)) + (dy)/(dz) =

Putting logx=z show that, x^2(d^2y)/(dx^2)=(d^2y)/(dz^2)-dy/dz .

If y=sin^(-1)x,z=cos^(-1)sqrt(1-x^(2))," then: "(dy)/(dz)=

If x=(1)/(z), y=f(x) and (d^(2)y)/(dx^(2))=kz^(3)(dy)/(dz)+z^(4)(d^(2)y)/(dz^(2)) , then the value of k is -

Solve the following differential equations: (dy)/(dx)=1+x+y+x y (ii) y-x(dy)/(dx)=a(y^2+(dy)/(dx))

Solve the following differential equations: (dy)/(dx)=1+x+y+x y (ii) y-x(dy)/(dx)=a(y^2+(dy)/(dx))

Solve the following differential equations: (dy)/(dx)=1+x+y+x y (ii) y-x(dy)/(dx)=a(y^2+(dy)/(dx))

"If "x^(y)+y^(x)+a^(x)+x^(a)=1," find "(dy)/(dx)

A: If y = x ^(y) then (dy)/(dx) = (y ^(2))/(x(1- log y )) If y = f (x) ^(y), then (dy)/(dx) = (y ^(2) f '(x))/(f (x) [1- ylog f (x)])= (y ^(2) f'(x))/(f (x) [1- log y])