You can also use Logarithmic Differentiation.
By #tan x=sin x/cos x#,
#y=(sin^6x cdot tan^2 x)/(x^2+2)^2
=(sin^6x cdot (sin^2 x)/(cos^2 x))/(x^2+2)^2
=(sin^8 x)/(cos^2x(x^2+2)^2)#
By taking the natural log of both sides,
#Rightarrow ln y=ln((sin^8 x)/(cos^2 x(x^2+2)^2))#
By Log Properties: #ln(x cdot y)=ln x + ln y# and #ln(x/y)=ln x - ln y,#
#Rightarrow ln y=ln(sin^8 x)-ln(cos^2 x)-ln(x^2+2)^2#
By Log Property: #ln x^r=r ln x#,
#Rightarrow ln y=8ln(sin x)-2ln(cos x)-2ln(x^2+2)#
By differentiating with respect to #x# using #[ln(g(x))]'=(g'(x))/g(x)#,
#Rightarrow (y')/y=8 cos x/sin x-2(-sin x)/(cos x)-2 (2x)/(x^2+2)#
By cleaning up a bit,
#Rightarrow (y')/y=8 cot x+2tan x-(4x)/(x^2+2)#
By multiplying both sides by #y#,
#Rightarrow y'=(8 cot x+2tan x-(4x)/(x^2+2))(sin^6x cdot tan^2 x)/(x^2+2)^2#
I hope that this was clear.