(2) y = cos 3 θ {\displaystyle y=\cos ^{3}\theta } .
This is the same thing as y = ( cos θ ) 3 {\displaystyle y=(\cos \theta )^{3}} .
Let cos θ = v {\displaystyle \cos \theta =v} ; then y = v 3 {\displaystyle y=v^{3}} ; d y d v = 3 v 2 {\displaystyle {\dfrac {dy}{dv}}=3v^{2}} .
d v d θ = − sin θ . d y d θ = d y d v × d v d θ = − 3 cos 2 θ sin θ . {\displaystyle {\begin{aligned}{\frac {dv}{d\theta }}&=-\sin \theta .\\{\frac {dy}{d\theta }}&={\frac {dy}{dv}}\times {\frac {dv}{d\theta }}=-3\cos ^{2}\theta \sin \theta .\end{aligned}}}
(3) y = sin ( x + a ) {\displaystyle y=\sin(x+a)} .
Let x + a = v {\displaystyle x+a=v} ; then y = sin v {\displaystyle y=\sin v} .
d y d v = cos v ; d v d x = 1 {\displaystyle {\frac {dy}{dv}}=\cos v;\quad {\frac {dv}{dx}}=1\quad } and d y d x = cos ( x + a ) {\displaystyle \quad {\frac {dy}{dx}}=\cos(x+a)} .
(4) y = log ϵ sin θ {\displaystyle y=\log _{\epsilon }\sin \theta } .
Let sin θ = v {\displaystyle \sin \theta =v} ; y = log ϵ v {\displaystyle y=\log _{\epsilon }v} .
d y d v = 1 v ; d v d θ = cos θ ; d y d θ = 1 sin θ × cos θ = cot θ . {\displaystyle {\begin{aligned}{\frac {dy}{dv}}&={\frac {1}{v}};\quad {\frac {dv}{d\theta }}=\cos \theta ;\\{\frac {dy}{d\theta }}&={\frac {1}{\sin \theta }}\times \cos \theta =\cot \theta .\end{aligned}}} .
(5) y = cot θ = cos θ sin θ {\displaystyle y=\cot \theta ={\dfrac {\cos \theta }{\sin \theta }}} .
d y d θ = − sin 2 θ − cos 2 θ sin 2 θ = − ( 1 + cot 2 θ ) = − cosec 2 θ . {\displaystyle {\begin{aligned}{\frac {dy}{d\theta }}&={\frac {-\sin ^{2}\theta -\cos ^{2}\theta }{\sin ^{2}\theta }}\\&=-(1+\cot ^{2}\theta )=-{\text{cosec}}^{2}\theta .\end{aligned}}} .
(6) y = tan 3 θ {\displaystyle y=\tan 3\theta } .
Let 3 θ = v {\displaystyle 3\theta =v} ; y = tan v {\displaystyle y=\tan v} ; d y d v = sec 2 v {\displaystyle {\dfrac {dy}{dv}}=\sec ^{2}v} .
d v d θ = 3 ; d y d θ = 3 sec 2 3 θ {\displaystyle {\frac {dv}{d\theta }}=3;\quad {\frac {dy}{d\theta }}=3\sec ^{2}3\theta } .
