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$$y'=\lim_{h\rightarrow 0}\dfrac {f\left( x+h\right) -f\left( x\right) }{h}$$

So, assuming that ##y= e^{x},## can we prove, using first principle, that:

$$\dfrac{dy}{dx}\left( e^{x}\right) =e^x$$

Or is there other methods that are primarily used to do so? Just curious, because my working lead me

to the final end of:

$$y'=\lim_{h\rightarrow 0}\dfrac {e^{x}\left( e^{h}-1\right) }{h}$$