Differentiate the following functions:

Solution:

(i) $6 x^{5}+4 x^{3}-3 x^{2}+2 x-7$

Formula:-

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

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

$\frac{d}{d x}\left(6 x^{5}+4 x^{3}-3 x^{2}+2 x-7\right)=30 x^{5-1}+12 x^{3-1}-6 x^{2-1}+2 x^{1-1}+0$

$=30 x^{4}+12 x^{2}-6 x^{1}+2 x$

(ii) $5 x^{-3 / 2}+\frac{4}{\sqrt{x}}+\sqrt{x}-\frac{7}{x}$

Formula:-

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

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

$\frac{d}{d x}\left(5 x^{-3 / 2}+\frac{4}{\sqrt{x}}+\sqrt{x}-\frac{7}{x}\right)$

$=5 \times-\frac{3}{2} x^{-\frac{3}{2}-1}+4 \times-\frac{1}{2} x^{-\frac{1}{2}-1}+\frac{1}{2} x^{\frac{1}{2}-1}-7 \times-1 x^{-1-1}$

$=-\frac{15}{2} x^{-\frac{5}{2}}-2 x^{-\frac{3}{2}}+\frac{1}{2} x^{\frac{-1}{2}}+7 x^{-2}$

(iii) $a x^{3}+b x^{2}+c x+d$, where $a, b, c, d$ are constants

Formula:-

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

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

$\frac{d}{d x}\left(a x^{3}+b x^{2}+c x+d\right)=3 a x^{3-1}+2 b x^{2-1}+c x^{1-1}+d \times 0$

$=3 a x^{2}+2 b x+c$