Suppose that $f(x, y)=x y$. The directional derivative of $f(x, y)$ in the directional $\langle 2,-6\rangle$ and at the point $(x, y)=(-1,-5)$ is

Step 1 ：Given the function \(f(x, y) = xy\), we want to find the directional derivative at the point \((-1, -5)\) in the direction of the vector \(\langle 2,-6\rangle\).

Step 2 ：The formula for the directional derivative of a function \(f(x, y)\) at a point \((x_0, y_0)\) in the direction of a vector \(\langle a, b \rangle\) is \(D_{\vec{v}}f(x_0, y_0) = f_x(x_0, y_0) \cdot a + f_y(x_0, y_0) \cdot b\), where \(f_x\) and \(f_y\) are the partial derivatives of \(f\) with respect to \(x\) and \(y\) respectively.

Step 3 ：For the function \(f(x, y) = xy\), the partial derivatives are \(f_x = y\) and \(f_y = x\).

Step 4 ：At the point \((-1, -5)\), we have \(f_x = -5\) and \(f_y = -1\).

Step 5 ：The direction vector is \(\langle 2, -6 \rangle\), so \(a = 2\) and \(b = -6\).

Step 6 ：Substituting these values into the formula, we get \(D_{\vec{v}}f(-1, -5) = -5 \cdot 2 + -1 \cdot -6\).

Step 7 ：Simplifying this expression, we find that the directional derivative of \(f(x, y)=x y\) in the direction \(\langle 2,-6\rangle\) and at the point \((x, y)=(-1,-5)\) is \(\boxed{-4}\).