Differentiate the following functions from first principles: e-x. - Vidyadip

Question

Differentiate the following functions from first principles:
e^-x.

✓

Answer

Consider f(x) = e^-x
⇒ f(x + h) = e^-(x+h)
$\frac{\text{d}}{\text{dx}}(\text{f}(\text{x}))=\frac{\lim}{\text{h}\rightarrow0}\frac{\text{f}(\text{x}+\text{h})-\text{f}(\text{x})}{\text{h}}$
$=\frac{\lim}{\text{h}\rightarrow0}\frac{\text{e}^{-(\text{x}+\text{h})}\text{e}^{-\text{x}}}{\text{h}}$
$=\frac{\lim}{\text{h}\rightarrow0}\frac{\text{e}^{-\text{x}}\times\text{e}^{-\text{h}}-\text{e}^{-\text{x}}}{\text{h}}$
$=\frac{\lim}{\text{h}\rightarrow0}\text{e}^{-\text{x}}\left\{\Big(\frac{\text{e}^{-\text{h}}-1}{-\text{h}}\Big)\right\}\times(-1)$
$\Big[\text{Since, }\frac{\lim}{\text{h}\rightarrow0}\frac{\text{e}^\text{h}-1}{\text{h}}=1\Big]$
$=-\text{e}^{-\text{x}}$
So,
$\frac{\text{d}}{\text{dx}}(\text{e}^{-\text{x}})=-\text{e}^{-\text{x}}$

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