1. Find the derivative of:
\begin{aligned}
y \ = \ {\Big(3x \ – \ 1 \Big)}^{3} \ \cos \frac{3}{2}x \ \sin \frac{3}{2}x .
\end{aligned}
2. Find the derivative of:
\begin{aligned}
y \ = \ 2 \ {\csc}^{3} \Big(4\pi {x}^{3} \ – \ \frac{\pi}{2} \Big).
\end{aligned}
3. Find the derivative of:
\begin{aligned}
y \ = \ \sqrt[3]{ 2 \ – \ \cot (1 \ – \ \sqrt{x}) }.
\end{aligned}
4. Find the derivative of:
\begin{aligned}
\hspace{2em} y \ = \ \frac{ \tan x – \cot x }{ \cot x + \tan x }.
\end{aligned}
5. Find the derivative of:
\begin{aligned}
\hspace{2em} y \ = \ \frac{ \sin t + \cos t }{ \sin t – \cos t }.
\end{aligned}
6. Find the derivative of:
\begin{aligned}
\hspace{2em} y \ = \ \ \frac{ \sin 2t + \sin 5t \ – \ \sin t }{ \cos 2t + \cos 5t + \cos t }.
\end{aligned}
7. If $ f(x) = -5 \sec 3x \ $ then find the value of $ f'(\frac{4 \pi}{9})$.
8. Find the equation of the tangent line and the equation of normal line to the curve $ y = \dfrac{-3}{\cot 2x} \ $ at point where $ x = \frac{\pi}{3}. $
9. Given that $ y = x^2 + 8x + 13 $
(i) find $ \frac{dy}{dx} $ and the value of $x$ for which $\frac{dy}{dx} {\small = 0} $
(ii) showing your working clearly, decide whether the point corresponding to this $x$ value is a maximum or minimum by considering the gradient either side of it
(iii) show that the corresponding $y$ value is -3
(iv) sketch the curve.
10. Find $\frac{dy}{dx}$ when $ y = x^3 \ – \ x \ $ and show that $ y = x^3 \ – \ x \ $ is an increasing function for $ x < -\frac{1}{\sqrt{3}} \ $ and $ \ x > \frac{1}{\sqrt{3}}.$