Differentiate the following functions from first principles:

We have to find the derivative of $\mathrm{e}^{3 x}$ with the first principle method, so,

$f(x)=e^{3 x}$

by using the first principle formula, we get,

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

$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{e^{3(x+h)}-e^{3 x}}{h}$

$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{e^{2 x}\left(e^{2 h}-1\right)}{h}$

$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{e^{3 x}\left(e^{3 h}-1\right) 3}{3 h}$

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

$f^{\prime}(x)=3 e^{3 x}$