Find the derivative of the following functions:(i) `sin x cos x` (ii) `sec x` (iii) `5 sec x + 4 cos x`(iv) `cos e c x` (v) `3 cot x + 5 cos e c x`

Explanation

(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`.