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Question
Differentiate the following functions from first principles e3x.
Solution
\[\text{ Let } f\left( x \right) = e^{3x} \]
\[ \Rightarrow f\left( x + h \right) = e^{3\left( x + h \right)} \]
\[\frac{d}{dx}\left( f\left( x \right) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{e^{3\left( x + h \right)} - e^{3x}}{h}\]
\[ = \lim_{h \to 0} \frac{e^{3x} e^{3h} - e^{3x}}{h}\]
\[ = \lim_{h \to 0} e^{3x} \left\{ \frac{\left( e^{3h} - 1 \right)}{3h} \right\} \times 3\]
\[ = 3 e^{3x} \lim_{h \to 0} \left\{ \frac{\left( e^{3h} - 1 \right)}{3h} \right\}\]
\[ = 3 e^{3x} \]
\[\text{ Hence }, \frac{d}{dx}\left( e^{3x} \right) = 3 e^{3x}\]
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