निम्नलिखित फलनों के अवकलज ज्ञात कीजिए ।
(i) `sin xcos x`
(ii) `secx`
(iii) `5 sec x+4cosx`
(iv) `cosec x`
(v) `3cot x+5cosec x`
(vi) `5 sin x-6cos x+7`
(vii) `2tanx-7sec x`

Explanation

(i) माना `y=sinx.cos x`
`rArr (dy)/(dx)=(d)/(dx)(sinx.cosx)`
`=sinx.(d)/(dx)cos x+cos x(d)/(dx)sinx`
`sinx(-sinx)+cosx.(cosx)`
`-= sin^2x+cos ^2x=cos2x`
(ii) माना `y=sec x=(1)/(cos x)`
`rArr (dy)/(dx)=(d)/(dx)((1)/(cosx))`
`=(cos x(d)/(dx)(1)-1.(d)/(dx)cosx)/(cos^2x)`
`=(0-(-sinx))/(cos^2x)`
`=(1)/(cosx).(sinx)/(cos x)=sec x tan x`
(iii) माना `y=5sec x+4cosx`
`rArr (dy)/(dx)=(d)/(dx)(5secx+4cosx)`
`=5sec x tan x-4sinx`
(iv) माना `y=cosec x =(1)/(sinx)rArr (dy)/(dx)=(d)/(dx)((1)/(sinx))`
`=(sinx(d)/(dx)(1)-1.(d)/(dx)sinx)/((sinx)^2)`
`=(0-cosx)/(sin^2x)=-(1)/(sinx).(cosx)/(sinx)`
`=-cosec xcot x`
(v) माना `y=3cotx+5cosecx`
`rArr (dy)/(dx)=(d)/(dx)(3cotx+5cosec x)`
`=3(d)/(dx)cotx+5(d)/(dx)cosecx`
`=-3cosec^2x-5cosec xcot x`
(vi) माना `y=5sinx-6cos x+7`
`rArr (dy)/(dx)=(d)/(dx)(5sinx-6cosx+7)`
`=5(d)/(dx)(sinx)-6(d)/(dx)(cosx)+(d)/(dx)(7)`
`=5cosx+6sinx+0`
`=5cosx+6sinx`
(vii) माना `y=2tanx-7secx`
`rArr(dy)/(dx)=(d)/(dx)(2tanx-7secx)`
`=2(d)/(dx)tanx-7(d)/(dx)secx`
`=2sec^2x-7sec x tan x`