(6) dy/dx=-cot(x)csc(x)+csc²(x)(cot(x)csc(x))=
cot(x)csc(x)(-1+csc²(x))=cot³(x)csc(x).
(7) dy/dx=(3sec²(2x))(2sec(2x)tan(2x))-6sec²(2x)tan(2x)=
6sec³(2x)tan(2x)-6sec²(2x)tan(2x)=
6sec²(2x)tan(2x)(sec(x)-1).
(8) dy/dx=⅜+⅜(-sin²(x)+cos²(x))+¼(-3cos²(x)sin²(x)+cos⁴(x))=
⅜+⅜cos(2x)+¼cos²(x)(cos²(x)-3sin²(x))=
⅜+⅜cos(2x)+¼cos²(x)(cos(2x)-2sin²(x))=
⅜+⅜cos(2x)+¼cos²(x)(2cos(2x)-1)=
⅜+⅜cos(2x)+⅛(cos(2x)+1)(2cos(2x)-1)=⅜+⅜cos(2x)+⅛(2cos²(2x)+cos(2x)-1)=
⅜+⅜cos(2x)+¼cos²(2x)+⅛cos(2x)-⅛=
¼+¼cos²(2x)+½cos(2x)=¼(cos(2x)+1)².
(9) dy/dx=3cos(x/3)-xsin(x/3)-3cos(x/3)=-xsin(x/3).
(10) dy/dx=4sec³(x)sec(x)tan(x)-4tan(x)sec²(x)
4sec⁴(x)tan(x)-4sec²(x)tan(x)=
4sec²(x)tan(x)(sec²(x)-1)=4sec²xtan³(x).
As with (1) to (5) the expression for y can be rewritten and then differentiated with the same results.