${(x + 3)^2} \cdot {(x + 4)^3} \cdot {(x + 5)^4}$

Let $y = {(x + 3)^2} \cdot {(x + 4)^3} \cdot {(x + 5)^4}$

By taking log on both sides , we get

$\log y = \log [{(x + 3)^2}{(x + 4)^3}{(x + 5)^4}]$
$\Rightarrow$ $\log y = 2\log (x + 3) + 3\log (x + 4) + 4\log (x + 5)$ …(i)

Differentiating (i) on both sides w.r.t. x, we get

$\Rightarrow$ $\cfrac{1}{y}\cfrac{{dy}}{{dx}} = 2 \cdot \cfrac{1}{{(x + 3)}} + 3 \cdot \cfrac{1}{{(x + 4)}} + 4 \cdot \cfrac{1}{{(x + 5)}}$

$\Rightarrow$ $\cfrac{{dy}}{{dx}} = {(x + 3)^2} \cdot {(x + 4)^3} \cdot {(x + 4)^4}\left[ {\cfrac{2}{{x + 3}} + \cfrac{3}{{x + 4}} + \cfrac{4}{{x + 5}}} \right]$

$= (x + 3){(x + 4)^2}{(x + 5)^3}(9{x^2} + 70x + 133)$