The directional derivative is just what its name suggests. It is the derivative of a function in a particular direction. The following picture illustrates the situation in the case of a function of two variables.
In this picture, v ≡
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It tells how fast z is changing in this direction. If you looked at it from the side, you would be getting the slope of the indicated tangent line. A simple example of this is a person climbing a mountain. He could go various directions, some steeper than others. The directional derivative is just a measure of the steepness in a given direction. This motivates the following general definition of the directional derivative.
Definition 11.3.1 Let f : U → ℝ where U is an open set in ℝn and let v be a unit vector. For x ∈ U, define the directional derivative of f in the direction v, at the point x as
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Example 11.3.2 Find the directional derivative of the function f
First you need a unit vector which has the same direction as the given vector. This unit vector is v ≡
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and to find the directional derivative, you take the limit of this as t → 0. However, this difference quotient equals
There is something you must keep in mind about this. The direction vector must always be a unit vector1 .