Differentiate the following functions from first principles :

We have to find the derivative of $\mathrm{e}^{-\mathrm{x}}$ with the first principle method, so,

$f(x)=e^{-x}$

by using the first principle formula, we get,

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

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

$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{e^{-x}\left(e^{-h}-1\right)}{h}$

$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{e^{-x}\left(e^{-h}-1\right)(-1)}{h(-1)}$

[By using $\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}=1$ ]

$f^{\prime}(x)=-e^{-x}$