Differentiate the functions with respect to 'x'. x^ 2 3

Question

Differentiate the functions with respect to 'x'.
$\text{x}^{\frac{2}{3}}$

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Answer

Let $\text{f}(\text{x})=\text{x}^{\frac{2}{3}}\ ...(\text{i})$
$\Rightarrow\text{f}(\text{x}+\Delta\text{x})=(\text{x}+\Delta\text{x})^{\frac{2}{3}}\ ...(\text{ii})$
Subtracting eq. (i) from eq. (ii)
$\Rightarrow\text{f}(\text{x}+\Delta\text{x}-\text{f}(\text{x})=(\text{x}+\Delta\text{x})^{\frac{2}{3}}-\text{x}^{\frac{2}{3}}$
Dividing both sides by taken the limit, we get
$=\lim\limits_{\Delta\text{x} \rightarrow 0}\frac{\text{f}(\text{x}+\Delta\text{x})-\text{f}(\text{x})}{\Delta\text{x}}$
$=\lim\limits_{\Delta\text{x} \rightarrow 0}\frac{(\text{x}+\Delta\text{x})^{\frac{2}{3}}}{\Delta\text{x}}$
$=\lim\limits_{\Delta\text{x} \rightarrow 0}\frac{\text{x}^{\frac{2}{3}}\Big[1+\frac{\Delta\text{x}}{\text{x}}\Big]^{\frac{2}{3}}-\text{x}^{\frac{2}{3}}}{\Delta\text{x}}$
$=\lim\limits_{\Delta\text{x} \rightarrow 0}\frac{\text{x}^{\frac{2}{3}}\Bigg[\bigg(1+\frac{\Delta\text{x}}{\text{x}}\bigg)^{\frac{2}{3}}\Bigg]-1}{\Delta\text{x}}$
$=\lim\limits_{\Delta\text{x} \rightarrow 0}\frac{\text{x}^{\frac{2}{3}}\bigg[\bigg(1+\frac{2}{3}\cdot\frac{\Delta\text{x}+\ ....}{\text{x}}\bigg)-1\bigg]}{\Delta\text{x}}$
$=\lim\limits_{\Delta\text{x} \rightarrow 0}\frac{\text{x}^{\frac{2}{3}}\cdot\frac{2}{3}\cdot\frac{\Delta\text{x}}{\text{x}}}{\Delta\text{x}}$
$=\frac{2}{3}\text{x}^{\frac{2}{3}-1}=\frac{2}{3}\text{x}^{\frac{-1}{3}}$
Hence, the required answer is $\frac{2}{3}\text{x}^{\frac{-1}{3}}.$