1.       Find the derivative of:
$$\begin{array}{r}y={(3x–1)}^3\cos \frac{3}{2}x\sin \frac{3}{2}x.\end{array}$$

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

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

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4.       Find the derivative of:
$$\begin{array}{r}y=\frac{\tan x–\cot x}{\cot x+\tan x}.\end{array}$$

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5.       Find the derivative of:
$$\begin{array}{r}y=\frac{\sin t+\cos t}{\sin t–\cos t}.\end{array}$$

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6.       Find the derivative of:
$$\begin{array}{r}y=\frac{\sin 2t+\sin 5t–\sin t}{\cos 2t+\cos 5t+\cos t}.\end{array}$$

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

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8.       Find the equation of the tangent line and the equation of normal line to the curve $y={\displaystyle \frac{-3}{\cot 2x}}$ 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 $\frac{dy}{dx}=0$
    (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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