Find the derivation of each of the following from the first principle:

Let $f(x)=x^{8}$

We need to find the derivative of f(x) i.e. f’(x)

We know that,

$\mathrm{f}^{\prime}(\mathrm{x})=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(\mathrm{x}+\mathrm{h})-\mathrm{f}(\mathrm{x})}{\mathrm{h}}$ …(i)

$f(x)=x^{8}$

$f(x+h)=(x+h)^{8}$

Putting values in (i), we get

$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{(x+h)^{8}-x^{8}}{h}$

$=\lim _{h \rightarrow 0} \frac{(x+h)^{8}-x^{8}}{(x+h)-x}$ 

[Add and subtract x in denominator]

$=\lim _{z \rightarrow x} \frac{z^{8}-x^{8}}{z-x}$ where $z=x+h$ and $z \rightarrow x$ as $h \rightarrow 0$

$=8 x^{8-1}\left[\because \lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}=n a^{n-1}\right]$

$=8 x^{7}$

Hence, $f^{\prime}(x)=8 x^{7}$