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172

Calculus Made Easy

(2) $y=\cos ^{3}\theta$ .

This is the same thing as $y=(\cos \theta )^{3}$ .

Let $\cos \theta =v$ ; then $y=v^{3}$ ; ${\dfrac {dy}{dv}}=3v^{2}$ .

${\begin{aligned}{\frac {dv}{d\theta }}&=-\sin \theta .\\{\frac {dy}{d\theta }}&={\frac {dy}{dv}}\times {\frac {dv}{d\theta }}=-3\cos ^{2}\theta \sin \theta .\end{aligned}}$

(3) $y=\sin(x+a)$ .

Let $x+a=v$ ; then $y=\sin v$ .

${\frac {dy}{dv}}=\cos v;\quad {\frac {dv}{dx}}=1\quad$ and $\quad {\frac {dy}{dx}}=\cos(x+a)$ .

(4) $y=\log _{\epsilon }\sin \theta$ .

Let $\sin \theta =v$ ; $y=\log _{\epsilon }v$ .

${\begin{aligned}{\frac {dy}{dv}}&={\frac {1}{v}};\quad {\frac {dv}{d\theta }}=\cos \theta ;\\{\frac {dy}{d\theta }}&={\frac {1}{\sin \theta }}\times \cos \theta =\cot \theta .\end{aligned}}$ .

(5) $y=\cot \theta ={\dfrac {\cos \theta }{\sin \theta }}$ .

${\begin{aligned}{\frac {dy}{d\theta }}&={\frac {-\sin ^{2}\theta -\cos ^{2}\theta }{\sin ^{2}\theta }}\\&=-(1+\cot ^{2}\theta )=-{\text{cosec}}^{2}\theta .\end{aligned}}$ .

(6) $y=\tan 3\theta$ .

Let $3\theta =v$ ; $y=\tan v$ ; ${\dfrac {dy}{dv}}=\sec ^{2}v$ .

${\frac {dv}{d\theta }}=3;\quad {\frac {dy}{d\theta }}=3\sec ^{2}3\theta$ .