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

Solution:

Let ,

$f(x)=\frac{1}{x^{5}}$

We need to find the derivative of $f(x)$ i.e. $f^{\prime}(x)$

We know that,

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

$f(x)=\frac{1}{x^{5}}$

$f(x+h)=\frac{1}{(x+h)^{5}}$

Putting values in (i), we get

$\mathrm{f}^{\prime}(\mathrm{x})=\lim _{\mathrm{h} \rightarrow 0} \frac{\frac{1}{(\mathrm{x}+\mathrm{h})^{5}}-\frac{1}{\mathrm{x}^{5}}}{\mathrm{~h}}$

$=\lim _{h \rightarrow 0} \frac{(x+h)^{-5}-x^{-5}}{(x+h)-x}$

[Add and subtract x in denominator]

$=\lim _{z \rightarrow x} \frac{z^{-5}-x^{-5}}{z-x}$ where $z=x+h$ and $z \rightarrow x$ as $h \rightarrow 0$

$=(-5) x^{-5-1}\left[\because \lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}=n a^{n-1}\right]$

$=-5 x^{-6}$

$=-\frac{5}{x^{6}}$

Hence,

$f^{\prime}(x)=-\frac{5}{x^{6}}$