Differentiate the following functions:
(i) $\frac{1}{x}$
(ii) $\frac{1}{\sqrt{\mathrm{x}}}$
(iii) $\frac{1}{\sqrt[3]{x}}$
(i) $\frac{1}{x}=x^{-1}$
Formula:-
$\frac{d}{d x} x^{n}=n x^{n-1}$
Differentiating w.r.t $\mathrm{x}$,
$\frac{d}{d x} x^{-1}=-1 x^{-1-1}$
$=-\mathrm{X}^{-2}$
(ii) $\frac{1}{\sqrt{x}}=x^{-\frac{1}{2}}$
Formula:-
$\frac{d}{d x} x^{n}=n x^{n-1}$
Differentiating w.r.t $x$,
$\frac{d}{d x} x^{\frac{-1}{2}}=\frac{-1}{2} x^{-\frac{1}{2}-1}$
$=\frac{-1}{2} x^{-\frac{3}{2}}$
(iii) $\frac{1}{\sqrt[3]{x}}=x^{\frac{-1}{3}}$
Formula:-
$\frac{d}{d x} x^{n}=n x^{n-1}$
Differentiating w.r.t $x$,
$\frac{d}{d x} x^{\frac{-1}{3}}=\frac{-1}{3} x^{\frac{-1}{3}-1}$
$=-\frac{1}{3} \mathrm{x}^{-\frac{4}{3}}$
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