9.5 Directional Derivative


We saw in the last section that for a function f of two variables x and y, the partial derivatives ∂z/∂x and ∂z/∂y give the slope of the tangent to the trace, or curve of intersection of the surface defined by z = f(x, y) and vertical planes that are, respectively, parallel to the x- and y-coordinates axes. Equivalently, we can think of the partial derivative ∂z/∂x as the rate of change of the function f in the direction given by the vector i, and ∂z/∂y as the rate of change of the function f in the j-direction. There is no reason to confine our attention to just two directions. In this section we shall see how to find the rate of change of a differentiable function in any direction. See FIGURE 9.5.1 ...

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