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Question
Differentiate the following functions from first principles e−x.
Solution
\[\text{ Let } f\left( x \right) = e^{- x} \]
\[ \Rightarrow f\left( x + h \right) = e^{- \left( x + h \right)} \]
\[ \frac{d}{dx}\left\{ f\left( x \right) \right\} = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{e^{- \left( x + h \right)} - e^{- x}}{h}\]
\[ = \lim_{h \to 0} \frac{e^{- x} \times e^{- h} - e^{- x}}{h}\]
\[ = \lim_{h \to 0} e^{- x} \left\{ \frac{\left( e^{- h} - 1 \right)}{- h} \right\} \times \left( - 1 \right)\]
\[ = - e^{- x} \lim_{h \to 0} \left\{ \frac{\left( e^{- h} - 1 \right)}{- h} \right\} \]
\[ = - e^{- x} \left[ \because \lim_{h \to 0} \frac{e^{- h} - 1}{- h} = 1 \right]\]
\[So, \frac{d}{dx}\left( e^{- x} \right) = - e^{- x}\]
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