If y is implicit differentiable function of x such that `y(x+y)^(2)=x` then `int(dx)/(x+3y)`

Statement I If y is a function of x such that y(x-y)^(2)= x, then int (dx)/(x-3y)=1/2 [ log (x-y)^(2)-1] Statement II int (dx)/(x-3y)=log (x-3y)+C

If y=x^(2)+2x+3, then : int((dx)/(dy))dx=

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) .

Evaluate differentiation of y with respect to x : y=x^(2)sinx

For x in x!=0, if y(x) differential function such that x int_(1)^(x)y(t)dt=(x+1)int_(1)^(x)ty(t)dt then y(x) equals: (where C is a constant.)

Evaluate differentiation of y with respect to x : y=x^(2)-sinx

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) .

Evaluate differentiation of y with respect to x: (2x+1)^(-3)

Evaluate differentiation of y with respect to x: ((1)/(3x-2))