Differentiate the following from first principle3^ x^2 - Vidyadip

Question

Differentiate the following from first principle$3^{\text{x}^2}$

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Answer

$\text{f}(\text{x})=3^{\text{x}^2}$$\text{f}'(\text{x})=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(\text{x}+\text{h})-\text{f}(\text{x})}{\text{h}}$
$\frac{\text{d}}{\text{dx}}\big(3^{\text{x}^2}\big)=\lim_\limits{\text{h}\rightarrow0}\frac{3^{(\text{x}+\text{h})^2}-3^{\text{x}^2}}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{3^{\text{x}^2+\text{2xh}+\text{h}^2}-3^{\text{x}^2}}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{3^{\text{x}^2}(3^{\text{x}^2+\text{2xh}+\text{h}^2-\text{x}^2}-1)}{\text{h}}\times\frac{(\text{h}+\text{2x})}{(\text{h}+\text{2h})}$
$=3^{\text{x}^2}\lim_\limits{\text{h}\rightarrow0}\frac{3^{\text{h}(\text{h}+\text{2x})}-1}{\text{h}(\text{h}+\text{2x})}\lim_\limits{\text{h}\rightarrow0}(\text{h}+\text{2x})$
$=3^{\text{x}^2}\log3(2\text{x})$
$=2\text{x}3^{\text{x}^2}\log3$

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