Differentiate `e^(2x)` with respect to `x` from first principles

Explanation

Let `y=f(x)=e^{2 x}` then


`frac{d y}{d x}=lim _{h rightarrow 0} frac{f(x+h)-f(x)}{h}=lim _{h rightarrow 0} frac{e^{2 x+2 h}-e^{2 x}}{h} `

`=lim _{h rightarrow 0} e^{2 x}[(frac{e^{2 h}-1}{2 h}right)right] cdot 2=e^{2 x} cdot 2 `

`therefore [frac{d}{d x}(e^{2 x}right)=2 e^{2 x}] .`