Differentiate the following functions with respect to x:
$x^{n} l o g x_{a}$

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Solution

We have,

$\frac{d}{d x} (x^{n} l o g x_{a})$

$= l o g x_{a} \frac{d}{d x} (x^{n}) + x^{n} \frac{d}{d x} (l o g x_{a})$ [Using product rule]

$= n x^{n - 1} . l o g x_{a} + \frac{x^{n}}{l o g a} . \frac{1}{x}$ $[∵ l o g x_{a} = \frac{l o g x}{l o g a}]$

$= x^{n - 1} [n . l o g x_{a} + \frac{1}{l o g a}]$

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