We have noted previously that the instantaneous rate of change of a function at the point will depend on the direction in which the independent variables are changing.
Consider the function . The graph of this function is shown below. At ,. As we can see by looking at the graph, as we move away from the origin along the positive -axis the value of is increasing, i.e. the rate of change of the function will be positive. However, if we move away from the origin along the positive -axis the value of is decreasing, i.e. the rate of change of the function will be negative.
In the case that the direction is parallel to the positive x-axis we already know that the slope is given by the partial derivative and in the case that the direction is parallel to the positive -axis the slope is given by . In this section we will look at the problem of finding the slope of the function if we move away from the point in any direction.
