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  • Differentiate each of the functions with respect to ‘x’ Differentiate using first principle X^2/3

Differentiate each of the functions with respect to ‘x’

Differentiate using first principle X^{\frac{2}{3}}

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\\ f^{'} \left( x \right) =\mathop{\lim }_{ \Delta x \rightarrow 0}\frac{ \left( x+ \Delta x \right) ^{\frac{2}{3}}-x^{\frac{2}{3}}}{ \Delta x} \\ \\ =\mathop{\lim }_{ \Delta x \rightarrow 0}\frac{x^{\frac{2}{3}} \left[ \left( 1+\frac{ \Delta x}{x} \right) ^{\frac{2}{3}}-1 \right] }{ \Delta x} \\ \\ =\mathop{\lim }_{ \Delta x \rightarrow 0}\frac{x^{\frac{2}{3}} \left[ \left( 1+\frac{2}{3}\frac{ \Delta x}{x}+ \ldots \right) -1 \right] }{ \Delta x} \\ \\

   Expanding by binomial theorem and rejecting the higher powers of \Delta x \: \: as\: \: \Delta x \rightarrow 0 \\ \\ 

 

   \\ =\mathop{\lim }_{ \Delta x \rightarrow 0}\frac{x^{\frac{2}{3}} \left( \frac{2}{3}\frac{ \Delta x}{x} \right) }{ \Delta x}=\mathop{\lim }_{ \Delta x \rightarrow 0} \left( \frac{2}{3}\frac{x^{\frac{2}{3}}}{x} \right) =\frac{2}{3}x^{-\frac{1}{3}}

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