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Differentiate the following functions:

Question:

Differentiate the following functions:

(i) $\frac{1}{x}$

(ii) $\frac{1}{\sqrt{\mathrm{x}}}$

(iii) $\frac{1}{\sqrt[3]{x}}$

 

Solution:

(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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