2f(x+y+1)(dy)/(dx)=1

if x=(1+t)/t^3 ,y=3/(2t^2)+2/t satisfies f(x)*{(dy)/(dx)}^3=1+(dy)/(dx) then f(x) is:

if x=(1+t)/t^3 ,y=3/(2t^2)+2/t satisfies f(x)*{(dy)/(dx)}^3=1+(dy)/(dx) then f(x) is:

if x=(1+t)/t^3 ,y=3/(2t^2)+2/t satisfies f(x)*{(dy)/(dx)}^3=1+(dy)/(dx) then f(x) is:

STATEMENT -1 : for the function y= f(x), f(x) ,({1+((dy)/dx)^(2)}^(3/2))/((d^(2)y)/(dx^(2))) = - ({1+ (dx/dy)^(2)}^(3/2))/((d^(2)x)/(dy^(2))) STATEMENT -2 : (dy)/(dx) = (1/(dx))/dy and (d^(2)y)/(dx^(2)) = d/dx (dy/(dx))

If y_(1)(x) and y_(2)(x) are two solutions of (dy)/(dx)+f(x)y=r(x), then y_(1)(x)+y_(2)(x) is solution of : (A) (dy)/(dx)+f(x)y=0 (B) (dy)/(dx)+2f(x)y=r(x)(C)(dy)/(dx)+f(x)y=2r(x)(D)(dy)/(dx)+2f(x)y=2r(x)

If y = f(x) and x = g(y), where g is the inverse of f, i.e., g = f^(-1) and if (dy)/(dx) and (dx)/(dy) both exist and (dx)/(dy) ne 0 , show that (dy)/(dx) = (1)/((dx//dy)) . Hence, (1) find (d)/(dx) (tan^(-1)x) (2) If y=sin^(-1)x, -1lexle1, -(pi)/(2)leyle(pi)/(2) , then show that (dy)/(dx)=(1)/(sqrt(1-x^(2))) where |x| lt 1 .

A : If y = sqrt( sinx + y ) then (dy)/(dx) = (cos x )/( 2y -1) R: If y = sqrt(f(x) + y) then (dy)/(dx) = (f'(x))/(2y -1).

if f'(x)=sqrt(2x^2-1) and y=f(x^2) then (dy)/(dx) at x=1 is:

if f'(x)=sqrt(2x^2-1) and y=f(x^2) then (dy)/(dx) at x=1 is: