Answer
$T'(z)=2^{z}\left(\ln z+\frac{1}{z\ln 2}\right)$
Work Step by Step
$$T(z)=2^{z}\log_{2}z$$
Using the product rule of the derivatives it follows:
$$T'(z)=(2^{z})'\log_{2}z+2^{z}(\log_{2}z)'$$
$$T'(z)=2^{z}\ln2(z)'\log_{2}z+2^{z}(\log_{2}z)'$$
$$T'(z)=2^{z}\ln2(z)'\log_{2}z+2^{z}\frac{1}{z\ln 2}$$
$$T'(z)=2^{z}\ln2(1)\log_{2}z+2^{z}\frac{1}{z\ln 2}$$
$$T'(z)=2^{z}\ln2\log_{2}z+2^{z}\frac{1}{z\ln 2}$$
$$T'(z)=2^{z}\left(\ln2\log_{2}z+\frac{1}{z\ln 2}\right)$$
$$T'(z)=2^{z}\left(\ln2\cdot\frac{\ln z}{\ln 2}+\frac{1}{z\ln 2}\right)$$
$$T'(z)=2^{z}\left(\ln z+\frac{1}{z\ln 2}\right)$$
