Find the derivative of the following functions: (i) `sin x cos x` (ii) `sec x` (iii) `5sec x+4 cos x` (iv) `cosec x` (v) `3 cot x+5 cosec x` (vi) `5sinx-6cosx+7` (vii) `2tanx-7secx`

Find the derivative of the following functions: (i) `sin x cos x` (ii)

1.	Find the derivative of the following functions: (i) `sin x cos x` (ii) `sec x` (iii) `5sec x+4 cos x` (iv) `cosec x` (v) `3 cot x+5 cosec x` (vi) `5sinx-6cosx+7` (vii) `2tanx-7secx`
Answer» (i) Let `y=sinx.cosx` `rArr(dy)/(dx)=(d)/(dx)(sinx.cosx)` `=sin x.(d)/(dx)cosx+cosx(d)/(dx)sinx` `=sinx(-sinx)+cos x.(cos x)` `=-sin^(2)x+cos^(2)x= cos 2 x` (ii) Let `y=sec x=(1)/(cos x)` `rArr(dy )/(dx)=(d)/(dx)((1)/(cos x))` `=(cos x(d)/(dx)(1)-1(d)/(dx)cos x)/(cos^(2)x)` `=(0-(-sinx))/(cos^(2)x)` `=(1)/(cos x).(sinx)/(cosx)=sec x tan x`. (iii) Let `y=cosec x=(1)/(sinx)rArr(dy)/(dx)=(d)/(dx)(5secx+4cosx )` `=5(d)/(dx)(sec x)+4(d)/(dx)(cos x)` `=5sec xtan x-4sinx` (iv) Let `y=cosec x=(1)/(sinx)rArr(dy)/(dx)=(d)/(dx)((1)/(sin x))` `=(sin x(d)/(dx)(1)-1.(d)/(dx)sinx)/((sin x)^(2))` `=(0-cosx)/(sin^(2)x)=-(1)/(sinx).(cosx)/(sinx)` `-cosec x cot x` (v) Let `y =3cot x+5 cosec x` `rArr(dy)/(dx)=(d)/(dx)(3 cot x+5cosec x)` `=3(d)/(dx)cot x+5(d)/(dx) cosec x` `=-3cosec^(2)x-5 cosec x cot x` (vi) Let `y=6 sin x-6cos x+7` `rArr(dy)/(dx)=(d)/(dx)(5sin x-6 cos x+7)` `=5(d)/(dx)(sinx)-6(d)/(dx)(cos x)+(d)/(dx)(7)` `=5cos x+6 sin x+0` `=5cos x +6sin x` (vii) Let `y=2 tan x-7 sec x` `rArr(dy)/(dx)=(d)/(dx)(2 tan x-7sec x)` `=2(d)/(dx)tan x-7(d)/(dx)secx` `=2sec^(2)x-7sec x tan x`.