Differentiation of Trigonometric Functions

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(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.

by Top Rated User (1.0m points)
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all good now

thank you very much sir

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