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