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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}

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2.       Find the derivative of:
\begin{aligned}
y \ = \ 2 \ {\csc}^{3} \Big(4\pi {x}^{3} \ – \ \frac{\pi}{2} \Big).
\end{aligned}

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3.       Find the derivative of:
\begin{aligned}
y \ = \ \sqrt[3]{ 2 \ – \ \cot (1 \ – \ \sqrt{x}) }.
\end{aligned}

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4.       Find the derivative of:
\begin{aligned}
\hspace{2em} y \ = \ \frac{ \tan x – \cot x }{ \cot x + \tan x }.
\end{aligned}

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5.       Find the derivative of:
\begin{aligned}
\hspace{2em} y \ = \ \frac{ \sin t + \cos t }{ \sin t – \cos t }.
\end{aligned}

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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}

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7.       If $f(x) = -5 \sec 3x \$ then find the value of $f'(\frac{4 \pi}{9})$.

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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 $x = \frac{\pi}{3}.$

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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.

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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 $\ x > \frac{1}{\sqrt{3}}.$

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