Answer
Vector opposite the gradient: $-(3 \mathbf{i}+2 \mathbf{j})=\mathbf{V}$
Unit vector: $-\frac{1}{\sqrt{13}}(\mathbf{i}+3 \mathbf{j})=\mathbf{v}$
Rate of change: $-\sqrt{13} . e^{6}=-\|\nabla f(2,3)\|$
Work Step by Step
$f$ increases more quickly in the positive direction of $\nabla f(x, y)$
\[
\begin{array}{l}
f_{x}(x, y)=y e^{x y} \\
f_{x}(2,3)=3 e^{6} \\
x e^{x y}=f_{y}(x, y) \\
2 e^{6}=f_{y}(2,3)
\end{array}
\]
\[
(3 \mathbf{i}+2 \mathbf{j})e^{6}=\nabla f(2,3)
\]
Vector opposite the gradient: $-(3 \mathbf{i}+2 \mathbf{j})=\mathbf{V}$
Unit vector: $-\frac{1}{\sqrt{13}}(\mathbf{i}+3 \mathbf{j})=\mathbf{v}$
Rate of change: $-\sqrt{13} . e^{6}=-\|\nabla f(2,3)\|$
