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Question
Differentiate each of the following from first principle:
\[3^{x^2}\]
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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( 3^{x^2} \right) = \lim_{h \to 0} \frac{3^\left( x + h \right)^2 - 3^{x^2}}{h}`
\[ = \lim_{h \to 0} \frac{3^{x^2 + 2xh + h^2} - 3^{x^2}}{h}\]
\[ = \lim_{h \to 0} \frac{3^{x^2} \left( 3^{x^2 + 2xh + h^2 - x^2} - 1 \right)}{h} \times \frac{\left( h + 2x \right)}{\left( h + 2x \right)}\]
\[ = 3^{x^2} \lim_{h \to 0} \frac{3^{h\left( h + 2x \right)} - 1}{h\left( h + 2x \right)} \lim_{h \to 0} \left( h + 2x \right)\]
\[ = 3^{x^2} \log 3 \left( 2x \right)\]
\[ = 2x 3^{x^2} \log 3\]
\[\]
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