Differentiate the following functions from first principles :

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

$f(x)=e^{a x+b}$

by using the first principle formula, we get,

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

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

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

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

$f^{\prime}(x)=a e^{a x+b}$