91Ó°ÊÓ

To find the direction of the gradient vector at a point, we first need to compute the partial derivatives of the function f(x, y) with respect to x and y: \[\frac{\partial f}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x_0+\Delta x, y_0) - f(x_0, y_0)}{\Delta x}\] \[\frac{\partial f}{\partial y} = \lim_{\Delta y \to 0} \frac{f(x_0, y_0+\Delta y) - f(x_0, y_0)}{\Delta y}\]

Next, we evaluate the partial derivatives of f(x, y) at the given point (x0, y0): \(\frac{\partial f}{\partial x}(x_0, y_0)\) and \(\frac{\partial f}{\partial y}(x_0, y_0)\).

Now that we have the partial derivatives, we can find the gradient vector, ∇f(x0, y0), at the given point (x0, y0): \[\nabla f(x_0, y_0) = \left(\frac{\partial f}{\partial x}(x_0, y_0), \frac{\partial f}{\partial y}(x_0, y_0)\right)\]

The direction of the gradient vector, ∇f(x0, y0), represents the direction in which the function f(x, y) has the greatest rate of increase at the point (x0, y0). This direction can be found by converting the gradient vector to a unit vector (a vector with length 1). This can be done using the following formula: \[\text{Unit vector in the direction of }\nabla f(x_0, y_0) = \frac{\nabla f(x_0, y_0)}{||\nabla f(x_0, y_0)||}\] Where \(||\nabla f(x_0, y_0)||\) is the magnitude of the gradient vector, given by: \[||\nabla f(x_0, y_0)|| = \sqrt{ \left(\frac{\partial f}{\partial x}(x_0, y_0)\right)^2 + \left(\frac{\partial f}{\partial y}(x_0, y_0)\right)^2}\] Once the unit vector is calculated, the direction of the gradient vector at the given point can be expressed in terms of angles in the coordinate system, or simply as the direction of an arrow from the point (x0, y0) to a new point (x0 + unit vector's x-component, y0 + unit vector's y-component).