### 9.5 Directional Derivative

## INTRODUCTION

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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