Differentiate the following functions from first principles:

e^ax+b

$$\begin{gathered} \mathrm{Let}f\left(x\right)=e^{ax+b} \\ \Rightarrow f\left(x+h\right)=e^{a\left(x+h\right)+b} \\ \therefore \frac{d}{dx}\left(f\left(x\right)\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{e^{a\left(x+h\right)+b}-e^{\left(ax+b\right)}}{h} \\ =\underset{h\to 0}{\lim }\frac{e^{ax+b}{e}^{ah}-e^{ax+b}}{h} \\ =\underset{h\to 0}{\lim }{e}^{ax+b}\left\{\frac{\left(e^{ah}-1\right)}{ah}\right\}\times a \\ =a{e}^{ax+b}\underset{h\to 0}{\lim }\left\{\frac{\left(e^{ah}-1\right)}{ah}\right\} \\ =a{e}^{ax+b} \\ \mathrm{So},\frac{d}{dx}\left(e^{ax+b}\right)=a{e}^{ax+b} \end{gathered}$$