N2006 P2 Q2

Given that \(z=\frac{x}{{{\left( {{x}^{2}}+32 \right)}^{\frac{1}{2}}}}\), show that \(\frac{\mathrm{d}z}{\mathrm{d}x}=\frac{32}{{{\left( {{x}^{2}}+32 \right)}^{\frac{3}{2}}}}\).[3]

Find the exact value of the area of the region bounded by the curve \(y=\frac{1}{{{\left( {{x}^{2}}+32 \right)}^{\frac{3}{2}}}}\), the \(x\)-axis and the lines \(x=2\) and \(x=7\).[3]