Find the derivative of $e^{x} x^{n} l o g_{a} x$

2 e^{x} x^{n - 1} {n l o g_{a} x + \frac{1}{l o g_{e} a} + x l o g_{a} x}

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3 e^{x} x^{n - 1} {n l o g_{a} x + x l o g_{a} x}

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e^{x} x^{n - 1} {n l o g_{a} x + \frac{1}{l o g_{e} a} + x l o g_{a} x}

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e^{x} x^{n - 1}

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Solution

The correct option is B $e^{x} x^{n - 1} {n l o g_{a} x + \frac{1}{l o g_{e} a} + x l o g_{a} x}$
$\frac{d}{d x} (e^{x} x^{n} {log}_{a} x)$
$= e^{x} (\frac{d}{d x} (x^{n} . {log}_{a} x)) + e^{x} x^{n} {log}_{a} x$
$= e^{x} (n x^{n - 1} {log}_{a} x + \frac{x^{n}}{x {log}_{e} a}) + e^{x} x^{n} {log}_{a} x$
$= e^{x} n x^{n - 1} {log}_{a} x + \frac{e^{x} . x^{n}}{x {log}_{e} a} + e^{x} x^{n} {log}_{a} x$
$= e^{x} x^{n - 1} (\frac{1}{{log}_{e} a} + n {log}_{a} x + x {log}_{a} x)$

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