The derivative follows from the fundamental theorem of calculus:
[tex]\displaystyle\frac{\mathrm d}{\mathrm dx}\int_c^{g(x)}f(t)\,\mathrm dt=f(g(x))\dfrac{\mathrm dg}{\mathrm dx}[/tex]
where [tex]c[/tex] is any constant in the domain of [tex]f[/tex].
We have
[tex]g(x)=x^3\implies\dfrac{\mathrm dg}{\mathrm dx}=3x^2[/tex]
so
[tex]\displaystyle\frac{\mathrm d}{\mathrm dx}\int_2^{x^3}\ln(t^2)\,\mathrm dt=3x^2\ln((x^3)^2)=18x^2\ln x[/tex]
(applying the property [tex]\ln a^b=b\ln a[/tex])
