(1) (i) ${\displaystyle {\dfrac {dy}{d\theta }}=A\cos \left(\theta -{\dfrac {\pi }{2}}\right)}$;

(ii) ${\displaystyle {\dfrac {dy}{d\theta }}=2\sin \theta \cos \theta =\sin 2\theta }$ and ${\displaystyle {\dfrac {dy}{d\theta }}=2\cos 2\theta }$;

(iii) ${\displaystyle {\dfrac {dy}{d\theta }}=3\sin ^{2}\theta \cos \theta }$ and ${\displaystyle {\dfrac {dy}{d\theta }}=3\cos 3\theta }$.

(2) ${\displaystyle \theta =45^{\circ }}$ or ${\displaystyle {\dfrac {\pi }{4}}}$ radians.

(3) ${\displaystyle {\dfrac {dy}{dt}}=-n\sin 2\pi nt}$.

(4) ${\displaystyle a^{x}\log _{\epsilon }a\cos a^{x}}$.

(5) ${\displaystyle {\dfrac {\cos x}{\sin x}}={\text{cotan}}\;x}$.

(6) ${\displaystyle 18.2\cos \left(x+26^{\circ }\right)}$.

(7) The slop is ${\displaystyle {\dfrac {dy}{d\theta }}=100\cos \left(\theta -15^{\circ }\right)}$, which is a maximum when ${\displaystyle (\theta -15^{\circ })=0}$ or ${\displaystyle \theta =15^{\circ }}$; the value of the slope being then ${\displaystyle =100}$. When ${\displaystyle \theta =75^{\circ }}$ the slope is ${\displaystyle 100\cos(75^{\circ }-15^{\circ })=100\cos 60^{\circ }=100\times {\frac {1}{2}}=50}$.

(8) ${\displaystyle \cos \theta \sin 2\theta +2\cos 2\theta \sin \theta =2\sin \theta \left(\cos ^{2}\theta +\cos 2\theta \right)}$

${\displaystyle =2\sin \theta \left(3\cos ^{2}\theta -1\right)}$.

(9) ${\displaystyle amn\theta ^{n-1}\tan ^{m-1}\left(\theta ^{n}\right)\sec ^{2}\theta ^{n}}$.

(10) ${\displaystyle \epsilon ^{x}\left(\sin ^{2}x+\sin 2x\right)}$; ${\displaystyle \epsilon ^{x}\left(\sin ^{2}x+2\sin 2x+2\cos 2x\right)}$.

(11) (i) ${\displaystyle {\dfrac {dy}{dx}}={\dfrac {ab}{\left(x+b\right)^{2}}}}$;

(ii) ${\displaystyle {\dfrac {a}{b}}\epsilon ^{-{\frac {x}{b}}}}$;

(iii) ${\displaystyle {\dfrac {1}{90^{\circ }}}\times {\dfrac {ab}{\left(b^{2}+x^{2}\right)}}}$.

(12) (i) ${\displaystyle {\dfrac {dy}{dx}}=\sec x\tan x}$;

(ii) ${\displaystyle {\dfrac {dy}{dx}}=-{\dfrac {1}{\sqrt {1-x^{2}}}}}$;

(iii) ${\displaystyle {\dfrac {dy}{dx}}={\dfrac {1}{1+x^{2}}}}$;

(iv) ${\displaystyle {\dfrac {dy}{dx}}={\dfrac {1}{x{\sqrt {x^{2}-1}}}}}$;

(v) ${\displaystyle {\dfrac {dy}{dx}}={\dfrac {{\sqrt {3\sec x}}\left(3\sec ^{2}x-1\right)}{2}}}$.

(13) ${\displaystyle {\dfrac {dy}{d\theta }}=4.6\left(2\theta +3\right)^{1.3}\cos \left(2\theta +3\right)^{2.3}}$.