Differentiate the following functions:

Question:

Differentiate the following functions:

(i) $3 x^{-5}$

(ii) $\frac{1}{5 \mathrm{x}}$

(iii) $6 . \sqrt[3]{x^{2}}$

 

Solution:

(i) $3 x^{-5}$

Formula:-

$\frac{d}{d x} x^{n}=n x^{n-1}$

Differentiating with respect to $x$,

$\frac{d}{d x} 3 x^{-5}=3(-5) x^{-5-1}$

$=-15 x^{-6}$

(ii) $1 / 5 x=\frac{1}{5} x^{-1}$

Formula:-

$\frac{d}{d x} x^{n}=n x^{n-1}$

Differentiating with respect to $\mathrm{X}$,

$\frac{1}{5} \frac{d}{d x} x^{-1}=\frac{-1}{5} x^{-1-1}$

$=-\frac{1}{5} x^{-2}$

(iii) 6. $\sqrt[3]{x^{2}}=6 x^{\frac{2}{3}}$

Formula:-

$\frac{d}{d x} x^{n}=n x^{n-1}$

Differentiating with respect to $\mathrm{x}$,

$\frac{\mathrm{d}}{\mathrm{dx}} 6 \mathrm{x}^{\frac{2}{3}}=6 \times \frac{2}{3} \mathrm{x}^{\frac{2}{3}-1}$

$=4 x^{-\frac{1}{3}}$

 

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