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`

If x and y are differentiable functions of t, then (dy)/(dx)=(dy//dt)/(dx//dt)," if "(dx)/(dt)ne0 .

Differential Equations of the type (dy)/(dx)=f(y)

Differential equations of the type (dy)/(dx)=f(x)

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(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=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) .

What is the order of the differential equation ? (dy)/(dx)+y=1/((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) .