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 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

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

Show that the function y = xe^(-x^(2)//2) satisfies the equation x(dy/dx)=(1-x^(2))y

(y-x(dy)/(dx))=3(1-x^(2)(dy)/(dx))

If y =f(u) is differentiable function of u, and u=g(x) is a differentiable function of x, then prove that y= f [g(x)] is a differentiable function of x and (dy)/(dx)=(dy)/(du)xx(du)/(dx) .

If y=sin^(-1) x , show that ((dy)/(dx))_(x=0)=1 .

Find the second order derivative of the following functions If y = cos^(-1)x show that (1 - x^2)(d^2y)/(dx^2) - x (dy)/(dx) = 0

If x=f(t), y=g(t) are differentiable functions of parameter 't' then prove that y is a differentiable function of 'x' and (dy)/(dx)=((dy)/(dt))/((dx)/(dt)),(dx)/(dt) ne 0 Hence find (dy)/(dx) if x=a cos t, y= a sin t.