Question

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

Answer 1

(i) Here f(x) = sin x cos x

$\therefore f^{\prime }\left(x\right)=\frac{d}{dx}\left(sinxcosx\right)$

$=sinx\frac{d}{dx}\left(cosx\right)+cosx\frac{d}{dx}\left(sinx\right)$

$=sinx\times (-sinx)+cosx\times cosx$

$=co{s}^{2}x-si{n}^{2}x=cos2x$

(ii) Here f(x) = sec x

$\therefore f^{\prime }\left(x\right)=\frac{d}{dx}\left(secx\right)=secxtanx$

(iii) Here $f\left(x\right)=5secx+4cosx$

$\therefore f^{\prime }\left(x\right)=\frac{d}{dx}(5secx+4cosx)$

$=5\frac{d}{dx}\left(secx\right)+4\frac{d}{dx}\left(cosx\right)$

$=5secxtanx-4sinx$

(iv) Here $f\left(x\right)=coescx$

$\therefore f^{\prime }\left(x\right)=\frac{d}{dx}\left(cosecx\right)=-cosecxcotx$.

(v) Here $f\left(x\right)=3cotx+5cosecx$

$\therefore f^{\prime }\left(x\right)=\frac{d}{dx}[3cotx+5cosecx]$

$=3\frac{d}{dx}\left(cotx\right)+5\frac{d}{dx}\left(cosecx\right)$

$=-3cose{c}^{2}x-5cosecxcotx$.

(vi) Here $f\left(x\right)=5sinx-6cosx+7$

$\therefore f^{\prime }\left(x\right)=\frac{d}{dx}[5sinx-6cosx+7]$

$=5\frac{d}{dx}\left(sinx\right)-6\frac{d}{dx}\left(cosx\right)+\frac{d}{dx}\left(7\right)$

$=5cosx+6sinx+0$

$=5cosx+6sinx$

(vii) Here f(x) = $2tanx-7secx$

$\therefore f^{\prime }\left(x\right)=\frac{d}{dx}[2tanx-7secx]$

$=2\frac{d}{dx}\left(tanx\right)-7\frac{d}{dx}\left(secx\right)$

$=2se{c}^{2}x-7secxtanx$