Start from the given #f (x)=[(sin x+ tan x)/(sin x cos x)]^3# and simplify it first
#f(x)=[sin x/(sin x cos x)+tan x/(sin x cos x)]^3#
#f(x)=[cancelsin x/(cancelsin x cos x)+(cancelsin x/cos x)*1/(cancelsin x cos x)]^3#
#f(x)=[1/( cos x)+(1/cos x)*1/(cos x)]^3#
#f(x)=[1/( cos x)+1/cos^2 x]^3#
#f(x)=[sec x+sec^2 x]^3#
At this point we differentiate using the power formula
#d/dx(u^n)=n*u^(n-1)*d/dx(u)#
#f(x)=[sec x+sec^2 x]^3#
#f' (x)=d/dx[sec x+sec^2 x]^3#
#f' (x)=3*[sec x+sec^2 x]^(3-1)*d/dx(sec x+sec^2 x)#
#f' (x)=3*[sec x+sec^2 x]^2*(sec x*tan x+2*(sec x)^(2-1)d/dx(sec x))#
#f' (x)=3*[sec x+sec^2 x]^2*(sec x*tan x+2*sec x*(sec x*tan x))#
#f' (x)=3*[sec x+sec^2 x]^2*(sec x*tan x+2*sec^2 x*tan x)#
#color(red)(f' (x)=3*sec^3 x*tan x*(1+sec x)^2(1+2*sec x))#
God bless...I hope the explanation is useful.
