#f(x)# can be written as #f(x)=(x+5)^2/(x^2+2)^2=u/v#
The quotient rule says that if #f(x)=g(x)/(h(x))#, then #f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2#
In this case:
#g(x)=(x+5)^2#
#g'(x)=[d/dx(x+5)*d/dx(u^2)],u=(x+5)#
#g'(x)=1*2u#
#g'(x)=1*2(x+5)#
#g'(x)=2(x+5)#
#h(x)=(x^2+2)^2#
#h'(x)=[d/dx(x^2+2)*d/dx(u^2)],u=(x^2+2)#
#h'(x)=2x*2u#
#h'(x)=2x*2(x^2+2)#
#h'(x)=4x(x^2+2)#
#f'(x)=((x^2+2)^(2)2(x+5)-(x+5)^(2)4x(x^2+2))/((x^2+2)^2)^2#
#f'(x)=(2(x^2+2)^2(x+5)-4x(x+5)^(2)(x^2+2))/(x^2+2)^4#
#f'(x)=(2(x^2+2)(x+5)-4x(x+5)^2)/(x^2+2)^3#
#f'(x)=(2(x+5)((x^2+2)-2x(x+5)))/(x^2+2)^3#
#f'(x)=(2(x+5)(-x^2-10x+2))/(x^2+2)^3#
#f'(x)=-(2(x+5)(x^2+10x-2))/(x^2+2)^3#
