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Question
x ex
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Solution
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( x e^x \right) = \lim_{h \to 0} \frac{(x + h ) e^{(x + h)} - x e^x}{h}\]
\[ = \lim_{h \to 0} \frac{(x + h) e^x e^h - x e^x}{h}\]
\[ = \lim_{h \to 0} \frac{x e^x e^h + h e^x e^h - x e^x}{h}\]
\[ = \lim_{h \to 0} \frac{x e^x e^h - x e^x}{h} + \lim_{h \to 0} \frac{h e^x e^h}{h}\]
\[ = \lim_{h \to 0} \frac{x e^x \left( e^h - 1 \right)}{h} + \lim_{h \to 0} e^x e^h \]
\[ = x e^x \left( 1 \right) + e^x \left( e^0 \right)\]
\[ = x e^x + e^x\]
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