Given function is
[tex]y=\ln \lbrack(x+3)^5(x+8)^4(x+4)^6\rbrack[/tex]
Now, using the property of logarithm, y can be expressed as
[tex]\begin{gathered} y=\ln (x+3)^5+\ln (x+8)^4+\ln (x+4)^6 \\ =3\ln (x+3)+4\ln (x+8)+6\ln (x+4) \end{gathered}[/tex]
Now, differentiating y w.r.t x,
[tex]\begin{gathered} \frac{d}{dx}(y)=\frac{3}{x+3}+\frac{4}{x+8}+\frac{6}{x+4}_{} \\ =\frac{13x^2+130x+288}{(x+3)(x+8)(x+4)} \end{gathered}[/tex]
So, the value of the derivative is
[tex]\frac{13x^2+130x+288}{(x+3)(x+8)(x+4)}[/tex]
