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Question
Differentiate each of the following from first principle:
e−x
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Solution
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( e^x \right) = \lim_{h \to 0} \frac{e^{- (x + h)} - e^{- x}}{h}\]
\[ = \lim_{h \to 0} \frac{e^{- x} e^{- h} - e^{- x}}{h}\]
\[ = \lim_{h \to 0} \frac{e^{- x} \left( e^{- h} - 1 \right)}{h}\]
\[ = - e^{- x} \lim_{h \to 0} \frac{e^{- h} - 1}{- h}\]
\[ = - e^{- x} \left( 1 \right)\]
\[ = - e^{- x}\]
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