${(x + 1)^2}{(x + 2)^3}{(x + 3)^4}$

Let $y = {(x + 1)^2}{(x + 2)^3}{(x + 3)^4}$
therefore,$\log y = \log \left\{ {{{(x + 1)}^2} \cdot {{(x + 2)}^3}{{(x + 3)}^4}} \right\}$

$= \log {(x + 1)^2} + \log {(x + 2)^3} + \log {(x + 3)^4}$
and $\frac{d}{{dy}}\log y \cdot \frac{{dy}}{{dx}} = \frac{d}{{dx}}[2\log (x + 1)] + \frac{d}{{dx}}[3\log (x + 2)] + \frac{d}{{dx}}[4\log (x + 3)]$

$\frac{1}{y} \cdot \frac{{dy}}{{dx}} = \frac{2}{{(x + 1)}} \cdot \frac{d}{{dx}}(x + 1) + 3 \cdot \frac{1}{{(x + 2)}} \cdot \frac{d}{{dx}}(x + 2) + 4 \cdot \frac{1}{{(x + 3)}} \cdot \frac{d}{{dx}}(x + 3)$

$= \left[ {\frac{2}{{x + 1}} + \frac{3}{{x + 2}} + \frac{4}{{x + 3}}} \right]$
therefore,$\frac{{dy}}{{dx}} = y\left[ {\frac{2}{{(x + 1)}} + \frac{3}{{(x + 2)}} + \frac{4}{{(x + 3)}}} \right]$

$= {(x + 1)^2} \cdot {(x + 2)^3} \cdot {(x + 3)^4}\left[ {\frac{2}{{(x + 1)}} + \frac{3}{{(x + 2)}} + \frac{4}{{(x + 3)}}} \right]$

$= {(x + 1)^2} \cdot {(x + 2)^3} \cdot {(x + 3)^4}$
$\left[ {\frac{{2(x + 2)(x + 3) + 3(x + 1)(x + 3) + 4(x + 1)(x + 2)}}{{(x + 1)(x + 2)(x + 3)}}} \right]$

$= \frac{{{{(x + 1)}^2}{{(x + 2)}^3}{{(x + 3)}^4}}}{{(x + 1)(x + 2)(x + 3)}}$

$\left[ {2\left( {{x^2} + 5x + 6} \right) + 3\left( {{x^2} + 4x + 3} \right) + 4\left( {{x^2} + 3x + 2} \right)} \right]$

$= (x + 1){(x + 2)^2}{(x + 3)^3}$

$\left[ {2{x^2} + 10x + 12 + 3{x^2} + 12x + 9 + 4{x^2} + 12x + 8} \right]$

$= (x + 1){(x + 2)^2}{(x + 3)^3}\left[ {9{x^2} + 34x + 29} \right]$

$\Rightarrow \frac{dy}{dx} == (x + 1){(x + 2)^2}{(x + 3)^3}\left[ {9{x^2} + 34x + 29} \right]$