Differentiate $e^{x}$ by first principle

e^{x}

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x e^{x}

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x e^{- x}

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- e^{x}

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Solution

The correct option is A $e^{x}$
$y = e^{x}$
$\frac{d y}{d x} = lim h \to 0 \frac{f (x + h) - f (x)}{h}$
$= lim h \to 0 \frac{e^{x + h} - e^{x}}{h}$
$= lim h \to 0 \frac{e^{x} (e^{h} - 1)}{h}$
$= e^{x}$
$∴ \frac{d y}{d x} = e^{x}$
Hence, option A.

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