Differentiate the following functions:

(i) $4 x^{3}+3 \cdot 2^{x}+6 \cdot \sqrt[8]{x^{-4}}+5 \cot x$

$=4 x^{3}+3 \cdot 2^{x}+6 x^{-\frac{1}{2}}+5 \cot x$

Formulae:

$\frac{d}{d x} x^{n}=n x^{n-1}$

$\frac{d}{d x} \cot x=-\operatorname{cosec}^{2} x$

$\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{a}^{\mathrm{x}}=\log _{\mathrm{n}}(\mathrm{a}) \times \mathrm{a}^{\mathrm{x}}$

Differentiating with respect to $\mathrm{x}$,

$\frac{d}{d x}\left(4 x^{3}+3 \cdot 2^{x}+6 x^{-\frac{1}{2}}+5 \cot x\right)$

$=4 \cdot 3 x^{3-1}+3 \cdot \log _{n}(2) \cdot 2^{x}+6 x-\frac{1}{2} x^{-\frac{1}{2}-1}+5 x-\operatorname{cosec}^{2} x$

$=12 x^{2}+3 \cdot \log _{n}(2) \cdot 2^{x}-3 x^{-\frac{3}{2}}-5 \operatorname{cosec}^{2} x$

(ii) $\frac{x}{3}-\frac{3}{x}+\sqrt{x}-\frac{1}{\sqrt{x}}+x^{2}-2^{x}+6 x^{-2 / 3}-\frac{2}{3} x^{6}$

$=\frac{x}{3}-3 x^{-1}+x^{\frac{1}{2}}-x^{-\frac{1}{2}}+x^{2}-2^{x}+6 x^{-2 / 3}-\frac{2}{3} x^{6}$

$\frac{d}{d x} x^{n}=n x^{n-1}$

$\frac{d}{d x} a^{x}=\log _{n}(a) \times a^{x}$

Differentiating with respect to $x$,

$\frac{d}{d x}\left(\frac{x}{3}-3 x^{-1}+x^{\frac{1}{2}}-x^{-\frac{1}{2}}+x^{2}-2^{x}+6 x^{-\frac{2}{3}}-\frac{2}{3} x^{6}\right)$

$\begin{aligned} & \frac{1}{3}-(-1) \times 3 x^{-1-1}+\frac{1}{2} x^{\frac{1}{2}-1}-\left(-\frac{1}{2}\right) x^{-\frac{1}{2}-1}+2 x^{2-1}-\log (2) \cdot 2^{x}+\\ & 6\left(-\frac{2}{3}\right) x^{-\frac{2}{3}-1}-\frac{2}{3} \times 6 x^{6-1} \end{aligned}$

$=\frac{1}{3}+3 x^{-2}+\frac{1}{2} x^{-\frac{1}{2}}+\frac{1}{2} x^{-\frac{3}{2}}+2 x^{1}-\log (2) \cdot 2^{x}-4 x^{-\frac{5}{3}}-4 x^{5}$