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 $\mathrm{x}$,
$\frac{\mathrm{d}}{\mathrm{dx}} 6 \mathrm{x}^{\frac{2}{3}}=6 \times \frac{2}{3} \mathrm{x}^{\frac{2}{3}-1}$
$=4 x^{-\frac{1}{3}}$