Differentiate the following function from first principles: `e^(3x)`

Explanation

`because f^{prime}(x) =lim _{h rightarrow 0} frac{f(n+h)-f(n)}{h} `

`because f(x) =e^{3 x} `

So, `f^{prime}(x) =lim _{h rightarrow 0} frac{e^{3(x+h)}-e^{3 n}}{h} `

`=lim _{h rightarrow 0} e^{3 n} frac{(e^{3 h}-1)}{h} `

`=3 cdot lim _{h rightarrow 0} frac{e^{3 n}(e^{3 h}-1)}{3 h} `

`=3 e^{3 x} lim _{3 h rightarrow 0} frac{(e^{3 h}-1)}{3 h} quad[because {h rightarrow 0} Rightarrow {3 h rightarrow 0}] `

`=3 cdot e^{3 x} cdot 1`

`=3 e^{3 x}`