Answer
$$0.74$$
Work Step by Step
We are given:
$$v_{\mathrm{avg}}=\sqrt{\frac{8 R T}{\pi M}}$$
Since $$ R= 8.31,\ \ \ \ M= 0.032 $$
Then
$$ v_{\mathrm{avg}}=\sqrt{\frac{8 R T}{\pi M}}= v_{\mathrm{avg}}=\sqrt{\frac{8 (8.31)T}{\pi (0.032)}} = \sqrt{\frac{2077.5}{\pi}}T^{1/2}$$
Hence
\begin{align*}
\frac{dv_{\mathrm{avg}}}{dT}&=\frac{1}{2} \sqrt{\frac{2077.5}{\pi}}T^{-1/2}\\
\frac{dv_{\mathrm{avg}}}{dT}\bigg|_{T=300}&=\frac{1}{2} \sqrt{\frac{2077.5}{\pi}}(300)^{-1/2}\\
&\approx 0.74
\end{align*}
