Find dy/dx and d^2y/dx^2 if y = x/(x^2+5)?

y=x/(x^2+5)
y=x(x^2+5)^-1
dy/dx=x(-1)(2x)(x^2+5)^-2 +(x^2+5)^-1
dy/dx=-2x^2 (x^2+5)^-2 +(x^2+5)^-1
d^2y/dx^2=-2[(x^2)(x^2+5)^-3 (2x)+(2x)(x^2+5)^-2]-(2x(x^2+5)^-2
d^2y/dx^2=-2[-4x^3 (x^2+5)^-3 +2x(x^2+5)^-2]-2x(x^2+5)^-2
d^2y/dx^2=(8x^3)(x^2+5)^-3 -(4x^2)(x^2+5)^-2 -(2x)(x^2+5)^-2
d^2y/dx^2=2x(x^2 +5)^-2 [4x(x^2+5)^-1 -2x-1]

Sol
y=x/(x^2+5)
x^2y+5y=x
x^2y’+2xy+5y’=1
(x^2+5)y’=1－2xy
=1－2x*x/(x^2+5)
=(5－x^2)/(x^2+5)
y'=(5－x^2)/(x^2+5)^2………………………………………
y’(x^2+5)^2=5－x^2
y’*2*(x^2+5)*(2x)+(x^2+5)y”=－2x
(5－x^2)/(x^2+5)^2*2*(x^2+5)*(2x)+(x^2+5)y”=－2x
4x(5－x^2)/(x^2+5)+(x^2+5)y”=－2x
(x^2+5)y”=－2x－4x(5－x^2)/(x^2+5)
=[－2x(x^2+5)－4x(5－x^2)]/(x^2+5)
=(－2x^3－10x－20x+4x^3)/(x^2+5)
=(2x^3－30x)/(x^2+5)
y”=(2x^3－30x)/(x^2+5)^2……………………………………