Differentiate the following functions with respect to x: 3^ x - Vidyadip

Question

Differentiate the following functions with respect to x:
$3^{\text{x}\log\text{x}}$

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Answer

Let $\text{y}=3^{\text{x}\log\text{x}}$
Differentiate it with respect to x we get,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big(3^{\text{x}\log\text{x}}\big)$
$=3^{\text{x}\log\text{x}}\times\log_\text{e}3\frac{\text{d}}{\text{dx}}(\text{x}\log\text{x})$
[Using chain rule]
$=3^{\text{x}\log\text{x}}\times\log_\text{e}3\Big[\text{x}\frac{\text{d}}{\text{dx}}(\log\text{x})+\log\text{x}\frac{\text{d}}{\text{dx}}(\text{x})\Big]$
$=3^{\text{x}\log\text{x}}\times\log_\text{e}3\Big[\frac{\text{x}}{\text{x}}+\log\text{x}\Big]$
$=3^{\text{x}\log\text{x}}\big[1+\log\text{x}\big]\times\log_\text{e}3$
So,
$\frac{\text{d}}{\text{dx}}\big(3^{\text{x}\log\text{x}}\big)=3^{\text{x}\log\text{x}}\big[1+\log\text{x}\big]\log_\text{e}3$

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