quotient rule:
#(u/v)' = (u'v  v'u)/(v^2)#
#u(x) = x^2 + 2x + 5#
power rule: #(x^n)' = nx^(n1)#
#u(x) = x^2 + 2x + 5#
#u'(x) = 2x^1 + 2x^0 = 2x + 2#
#v(x) = x + 1#
#v'(x) = 1x^0 = 1#
#u'v = (2x + 2) * (x + 1) = 2x^2 + 4x + 2#
#v'u = 1 * (x^2 + 2x + 5) = x^2 + 2x + 5#
#v^2 = (x+1)^2#
#(u'v  v'u) = (2x^2 + 4x + 2)  (x^2 + 2x + 5)#
#= 2x^2 + 4x + 2  x^2  2x  5#
#= x^2 + 2x  3#
#(u'v  v'u)/(v^2) = ((x^2+2x3))/(x+1)^2#
the first derivative of #y = (x^2+2x+5)/(x+1)# is #(x^2+2x3)/(x+1)^2#

the quotient rule #(u/v)' = (u'v  v'u)/(v^2)# can be used again
where #u(x) = x^2 + 2x 3#
and #v(x) = (x+1)^2, or x^2+2x+1#.
#u'(x) = 2x^1 + 2x^0 = 2x + 2, or 2(x+1)#
#v'(x) = 2x^1 + 2x^0 = 2x + 2, or 2(x+1)#
#u'v = (2(x+1)) * (x+1)^2 = 2(x+1)^3 = 2x^3 + 6x^2 + 6x + 2#
#v'u = (2x+2) * (x^2+2x3) = 2x^3 + 2x^2 + 4x^2 + 4x 6x  6#
#= 2x^3 + 6x^2  2x  6#
#u'v  v'u = (2x^3 + 6x^2 + 6x + 2)  (2x^3 + 6x^2  2x  6)#
#= 2x^3 + 6x^2 + 6x + 2  2x^3  6x^2 + 2x + 6#
#= 8x + 8#
#(u'v  v'u)/v^2 = (8x + 8)/((x+1)^4#
#8x + 8 = 8(x+1)#
#(8x+8)/(x+1)^4 = (8(x+1))/(x+1)^4#
#= 8/(x+1)^3#
the second derivative of #y = (x^2+2x+5)/(x+1)# is #8/(x+1)^3#