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Find the second order derivatives of the functions given in \( x^3 \log x \)

$\begin{array}{1 1} x(5-6\log x) \\3x^2\log x+x^2 \\ x(5+6\log x) \\ 3x^3\log x+x^2\end{array} $

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  • $y=f(x)$
  • $\large\frac{dy}{dx}$$=f'(x)$
  • $\large\frac{d^2y}{dx^2}=\frac{d}{dx}\big(\frac{dy}{dx}\big)$
  • $\large\frac{d}{dx}$$(\log x)=\large\frac{1}{x}$
Step 1:
Differentiating with respect to $x$
$\large\frac{dy}{dx}$$=3x^2.\log x+\large\frac{1}{x}$$.x^3$
$\quad\;=3x^2\log x+x^2$
Step 2:
$\large\frac{d^2y}{dx^2}=\frac{d}{dx}$$(x^2)+3\large\frac{d}{dx}$$(x^2\log x)$
$\qquad=2x+3(x^2.\large\frac{1}{x}$$+\log x.2x)$
$\qquad=2x+3(x+\log x.2x)$
$\qquad=2x+3x+6x\log x$
$\qquad=5x+6x\log x$
$\qquad=x(5+6\log x)$
answered May 10, 2013 by sreemathi.v
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