Question

Differentiate the following functions with respect to x:
$x^{n}log{x}_a$

Answer 1

We have,

$\frac{d}{dx}\left(x^{n}log{x}_a\right)$

$=log{x}_a\frac{d}{dx}\left(x^n\right)+x^n\frac{d}{dx}\left(log{x}_a\right)$ [Using product rule]

$=n{x}^{n-1}.log{x}_a+\frac{x^n}{loga}.\frac{1}{x}$ $[\because log{x}_a=\frac{logx}{loga}]$

$=x^{n-1}[n.log{x}_a+\frac{1}{loga}]$