Answer
$\frac{27}{32}$
Work Step by Step
Using the quotient rule, we have
$\frac{dw}{dz}=\frac{(\sqrt {z}+z)\frac{d}{dz}(z^{2})-(z^{2})\frac{d}{dz}(\sqrt {z}+z)}{(\sqrt {z}+z)^{2}}$
$\frac{(\sqrt {z}+z)2z-(z^{2})(\frac{1}{2\sqrt {z}}+1)}{(\sqrt {z}+z)^{2}}$
$=\frac{2z\sqrt {z}+2z^{2}-\frac{z\sqrt {z}}{2}-z^{2}}{(\sqrt {z}+z)^{2}}$
$=\frac{z^{2}+\frac{3z\sqrt {z}}{2}}{(\sqrt {z}+z)^{2}}$
$\frac{dw}{dz}|_{z=9}=\frac{9^{2}+\frac{3\times9\times\sqrt {9}}{2}}{(\sqrt {9}+9)^{2}}=\frac{243}{288}=\frac{27}{32}$
