If `y=x^(1/m),` show that `m=log_yx.`

If 2x=y^(1/m)+y^(-1/m) , show that (x^2-1)(d^2y)/(dx^2)+xdy/dx=m^2y .

If y={x+sqrt(x^2+1)}^m , show that (x^2+1)y_2+x y_1-m^2\ y=0

If y={x+sqrt(x^(2)+1)}^(m), show that (x^(2)+1)y_(2)+xy_(1)-m^(2)y=0

If y=cos(m sin^(-1)x), show that (1-x^(2))y_(2)-xy_(1)+m^(2)y=0

If (x-y)^(m+n)=x^m.y^n , show that dy/dx=y/x

If x^(m)*y^(n)=(x+y)^(m+n), show that (dy)/(dx)=(y)/(x)

If x^(y)=e^(x-y), then show that (dy)/(dx)=(log x)/((1+log x)^(2))

If x^y = e^(x-y) , then show that dy/dx = frac{log x}{(1+ log x)^2}

If x^y = e^(x + y) , show that (dy)/(dx) = (log x - 2)/((1 - log x)^2)

If 2 cos alpha = x + (1)/(x) and 2 cos beta = y + (1)/(y) , show that x^(m) y^(n) + (1)/(x^(m)y^(n)) = 2 cos (m alpha + n beta)