Find the second-order derivative of the function x^3 log x - Vidyadip

Question

Find the second-order derivative of the function $x^3$ log $x$

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Answer

Let $y = x^3$ log x$
$\therefore \frac{{dy}}{{dx}} = {x^3}\frac{d}{{dx}}\log x + \log x\frac{d}{{dx}}{x^3}$
$= {x^3}.\frac{1}{x} + \log x\left( {3{x^2}} \right)$
$= x^2 + 3x^2log x$
$\Rightarrow \frac{{{d^2}y}}{{d{x^2}}} = \frac{d}{{dx}}\left( {{x^2} + 3{x^2}\log x} \right)$
$= \frac{d}{{dx}}{x^2} + 3\frac{d}{{dx}}\left( {{x^2}\log x} \right)$
$= 2x + 3\left[ {{x^2}\frac{d}{{dx}}\log x + \log x\frac{d}{{dx}}{x^2}} \right]$
$= 2x + 3\left( {{x^2}.\frac{1}{x} + \left( {\log x} \right)2x} \right)$
$= 2x + 3(x + 2x \log x)$
$= 2x + 3x + 6x \log x$
$= 5x + 6x \log x$
$= x(5 + 6 \log x)$

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