Answer
$\displaystyle \frac{dy}{dx}=\frac{4x^{2}+9x+4}{2\sqrt{(x+1)(x+2)}}$
Work Step by Step
$y=\sqrt{x^{2}(x+1)(x+2)}$
$...$apply ln( ) to both sides
... on the RHS, apply $\ln M^{n}=n\ln M,$
$\displaystyle \ln y=\frac{1}{2}\ln[x^{2}(x+1)(x+2)]$
... on the RHS, apply $\ln(M\cdot N)=\ln M+\ln N$
$\displaystyle \ln y=\frac{1}{2}[2\ln(x)+\ln(x+1)+\ln(x+2)]\qquad $...$/\displaystyle \frac{d}{dx}$
$\displaystyle \frac{1}{y}(\frac{dy}{dx})=\frac{1}{2}[\frac{2}{x}+\frac{1}{x+1}+\frac{1}{x+2}]$
$\displaystyle \frac{1}{y}(\frac{dy}{dx})=\frac{1}{2}[\frac{2(x+1)(x+2)+x(x+2)+x(x+1)}{x(x+1)(x+2)}]$
$\displaystyle \frac{1}{y}(\frac{dy}{dx})=\frac{1}{2}[\frac{2(x^{2}+3x+2)+x^{2}+2x+x^{2}+x}{x(x+1)(x+2)}]$
$\displaystyle \frac{1}{y}(\frac{dy}{dx})=\frac{4x^{2}+9x+4}{2x(x+1)(x+2)} \qquad/\times y$
$... y=|x|(x+1)^{1/2}(x+2)^{1/2}=x(x+1)^{1/2}(x+2)^{1/2}$, since $x > 0$
$\displaystyle \frac{dy}{dx}=\frac{(4x^{2}+9x+4)\cdot y}{2x(x+1)(x+2)}$
$\displaystyle \frac{dy}{dx}=\frac{(4x^{2}+9x+4)x(x+1)^{1/2}(x+2)^{1/2}}{2x(x+1)(x+2)}$
$\displaystyle \frac{dy}{dx}=\frac{4x^{2}+9x+4}{2(x+1)^{1/2}(x+2)^{1/2}}$
$\displaystyle \frac{dy}{dx}=\frac{4x^{2}+9x+4}{2\sqrt{(x+1)(x+2)}}$
