x+y=cos(y/x) find d^2y/dx^2

Question

Differentiation implicit functions

Answer 1

x+y=cos(y/x).

Let z=y/x, so y=xz and dy/dx=xdz/dx+z; and x+xz=cosz. So -sinzdz/dx=1+z+xdz/dx and dz/dx=-(1+z)/(x+sinz).

Differentiating again: -coszd2z/dx2=0+dz/dx+xd2z/dx2+dz/dx=2dz/dx+xd2z/dx2⇒(substituting for cosz=x+xz)

-(x+xz)d2z/dx2=2dz/dx+xd2z/dx2⇒(2x+xz)d2z/dx2+2dz/dx=0 and d2z/dx2=-2dz/dx/(2x+xz). Substitute for dz/dx:

d2z/dx2=2(1+z)/(x(x+sinz)(2+z)).

d2y/dx2=d/dx(xdz/dx+z)=xd2z/dx2+2dz/dx=xd2z/dx2-2(1+z)/(x+sinz). Substitute for d2z/dx2:

d2y/dx2=2(1+z)/((x+sinz)(2+z))-2(1+z)/(x+sinz)=2(1+z)/((x+sinz)(2+z))(1-2-z)=

-2(1+z)^2/((x+sinz)(2+z)).

Now replace z with y/x:

d2y/dx2=-2(1+y/x)^2/((x+sin(y/x))(2+y/x))