The quotient rule is:
#(d(u/v))/dx = (u'v - uv')/v^2#
In this case, #u = x^2+x^2tan^2(x)# and #v = sec^2(x)#, then
#u' = 2x + 2xtan^2(x)+ 2x^2tan(x)sec^2(x)# and #v' = 2tan(x)sec^2(x)#
Substituting these values into the quotient rule:
#(d((x^2+x^2tan^2(x))/sec^2(x)))/dx = ((2x + 2xtan^2(x)+ 2x^2tan(x)sec^2(x))sec^2(x) - (x^2+x^2tan^2(x))(2tan(x)sec^2(x)))/sec^4(x)#
A common factor of #sec^2(x)/sec^2(x)# becomes 1:
#(d((x^2+x^2tan^2(x))/sec^2(x)))/dx = ((2x + 2xtan^2(x)+ 2x^2tan(x)sec^2(x))- (x^2+x^2tan^2(x))(2tan(x)))/sec^2(x)#
The term #- (x^2+x^2tan^2(x))(2tan(x))# can be written as #-2x^2tan(x)sec^2(x))#:
#(d((x^2+x^2tan^2(x))/sec^2(x)))/dx = ((2x + 2xtan^2(x)+ 2x^2tan(x)sec^2(x)) -2x^2tan(x)sec^2(x))/sec^2(x)#
The last two terms in the numerator sum to 0:
#(d((x^2+x^2tan^2(x))/sec^2(x)))/dx = (2x + 2xtan^2(x))/sec^2(x)#
Use the identity #1+ tan^2(x) = sec^2(x)#:
#(d((x^2+x^2tan^2(x))/sec^2(x)))/dx = 2x sec^2(x)/sec^2(x)#
#sec^2(x)/sec^2(x)# becomes 1:
#(d((x^2+x^2tan^2(x))/sec^2(x)))/dx = 2x#
It would have been better to avoid using the quotient rule by making the substitution, #1+ tan^2(x) = sec^2(x)#, at the start:
#(x^2+x^2tan^2(x))/sec^2(x) = x^2sec^2(x)/sec^2(x) = x^2#
#(d(x^2))/dx = 2x#
