Answer
\[\frac{7\sqrt 5}{10}\]
Work Step by Step
\[g(s,t)=s\sqrt{t}\]
Differentiate $g$ with respect to $s$ treating $t$ constant:
\[g_s(s,t)=\sqrt{t}\]
Differentiate $g$ with respect to $t$ treating $s$ constant:
\[g_t(s,t)=\frac{s}{2\sqrt{t}}\]
\[g_s(2,4)=\sqrt{4}=2\]
\[g_t(2,4)=\frac{2}{2\sqrt{4}}=\frac{1}{2}\]
\[\nabla g(2,4)=g_s(2,4)\hat{i}+g_t(2,4)\hat{j}\]
\[\nabla g(2,4)=2\hat{i}+\left(\frac{1}{2}\right)\hat{j}\]
\[\mathbf{v}=2\hat{i}-\hat{j}\]
\[|\mathbf{v}|=\sqrt{2^2+(-1)^2}=\sqrt{5}\]
\[\hat{\mathbf{v}}=\frac{\mathbf{v}}{|\mathbf v|}=\frac{2}{\sqrt{5}}\hat{i}-\frac{1}{\sqrt{5}}\hat{j}\]
Required derivative is given by:
\[D_{\mathbf v}g(2,4)=\nabla g(2,4)\cdot \hat{\mathbf v}\]
\[D_{\mathbf v}g(2,4)=\left[2\hat{i}+\left(\frac{1}{2}\right)\hat{j}\right]\cdot\left[\frac{2}{\sqrt{5}}\hat{i}-\frac{1}{\sqrt{5}}\hat{j}\right]\]
\[D_{\mathbf v}g(2,4)=\frac{4}{\sqrt{5}}-\frac{1}{2\sqrt{5}}=\frac{7\sqrt{5}}{10}\]
