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Applying rules of differentiation

Using the Basic Differentiation Rules

1. Differentiating \(y\) with respect to \(x\) gives:
\begin{eqnarray*} y &=& \frac{x^\frac{3}{2}}{4} - 3\sqrt{x} +8 \\ \frac{\mathrm{d}y}{\mathrm{d}x} &=& \frac{1}{4} \times \frac{3}{2}x^{\frac{3}{2} - 1} - 3\times \frac{1}{2}x^{\frac{1}{2} - 1} + 0\\ &=& \frac{1}{4} \times \frac{3}{2}x^\frac{1}{2} - 3\times \frac{1}{2 } x^{-\frac{1}{2}}\\ &=& \frac{3\sqrt{x}}{8} - \frac{3}{2\sqrt{x}} \end{eqnarray*} 2. Differentiating with respect to \(x\) gives:
\begin{eqnarray*}  
&&\frac{\mathrm{d}}{\mathrm{d}x} \left(5x - x^2 + 3x^3 + 2 \ln x\right) \\  
&=& 5 \times 1 x^{1-1} - 2x^{2-1} + 3\times 3x^{3-1}+2\times \frac{1}{x} \\  
&=& 5 -2x +9x^2+\frac{2}{x}
\end{eqnarray*}


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