If y = f(x) is a derivable function of x such that the inverse function `x = f^(-1)(y)` is defined, then show that `(dx)/(dy)=(1)/((dy//dx))`, where `(dy)/(dx) ne 0`.

If y=f(x) is a differentiable function x such that invrse function x=f^(-1) y exists, then prove that x is a differentiable function of y and (dx)/(dy)=(1)/((dy)/(dx)) where (dy)/(dx) ne0 Hence, find (d)/(dx)(tan^(-1)x) .

If y = f(u) is a differentiable functions of u and u = g(x) is a differentiable functions of x such that the composite functions y = f[g(x) ] is a differentiable functions of x then (dy)/(dx) = (dy)/(du) . (du)/(dx) Hence find (dy)/(dx) if y = sin ^(2)x

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

If y=x+(1)/(x+y)," then: "(dy)/(dx)=

y-x(dy)/(dx)=x+y(dy)/(dx)

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

If x = f(t) and y = g(t) are differentiable functions of t so that y is a differentiable functions of x and if dx//dt ne 0 then (dy)/(dx) =(dy)/((dt)/((dx)/(dt)) . Hence find dy/dx if x= 2sint and y=cos2t

1+(dy)/(dx)=cos(x+y), where x+y=u

1+(dy)/(dx)=cos(x+y), where x+y=u

Suppose y=f (x) is differentiable functions of x on an interval I and y is one - one, onto and (dy) / (dx) ne 0 on I . Also if f^(-1)(y) is differentiable on f(I) then prove that (dx)/(dy) =(1)/(dy//dx), dy/dx ne 0