If `x=a(t+1/t)` and `y=a(t-1/t)` , prove that `(dy)/(dx)=x/y`

If x=f(t) and y=phi(t), prove that (d^2y)/(dx^2)=(f_1phi_2-f_2phi_1)/(f_1^3) where suffixes denote differentiation w.r.t.t.

If x=cost andy=sint , then prove that (dy)/(dx)=(1)/(sqrt3), at t=(2pi)/(3) .

If x=sin^(-1)((2t)/(1+t^(2)))andy=tan^(-1)((2t)/(1-t^(2))), then prove that (dy)/(dx)=1 .

Find (dy)/(dx) : x= e^(cos 2t) and y= e^(sin 2t) show that, (dy)/(dx)= (-y log x)/(x log y)

If x=sqrt(a^sin^((-1)t)), y=sqrt(a^cos^((-1)t)) , show that (dy)/(dx)=-y/x

If x = sqrt(a^(sin^(-1)t)), y= sqrt(a^(cos^(-1)t)) , show that (dy)/(dx)= -(y)/(x)

If log (x^2+y^2)=2t a n^(-1)\ (y/x), then show that (dy)/(dx)=(x+y)/(x-y)

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

If x=e^(-t^(2)), y=tan^(-1)(2t+1) , then (dy)/(dx)=