My solutions say: $$ \frac{\mathrm d}{\mathrm ds}F(s) = \frac{\mathrm {d}}{\mathrm {d}s}\int_0^\infty \! x^{s-1}f(x) \, \mathrm {d}x = \int_0^\infty \! \log(x)x^{s-1}f(x)\,\mathrm {d}x = M\left[(\log(x))f(x)\right] $$
I cant see how $$ \frac{\mathrm {d}}{\mathrm {d}s}\int_0^\infty \! x^{s-1}f(x) \, \mathrm {d}x = \int_0^\infty \! \log(x)x^{s-1}f(x)\, \mathrm {d}x $$