# Directional Derivatives Examples 4

Recall from the Directional Derivatives page that for a two variable real-valued function $z = f(x, y)$, the directional derivative of $f$ at a point $(x, y) \in D(f)$ in the direction of the unit vector $\vec{u} = (a, b)$ is given by the formula:

(1)

For a three variable real-valued function $w = f(x, y, z)$, the directional derivative of $f$ at a point $(x, y, z) \in D(f)$ in the direction of the unit vector $\vec{u} = (a, b, c)$ is given by the formula:

(2)

## Example 1

**Find the directional derivative of $f(x, y, z) = 2xe^y\cos z$ in the direction of the vector $(1, 2, 3)$.**

We must first find a vector that points in the same direction as $(1, 2, 3)$ and has norm $1$. Let $\vec{u}$ be this unit vector. Then:

(3)

Now applying the formula above and we have that:

(4)

## Example 2

**Find the directional derivative of $f(x, y, z, w) = xy^2z^3w^4$ in the direction of the vector $(-1, -2, -3, -4)$.**

We first need a unit vector that points in the direction of $(-1, -2, -3, -4)$. Let $\vec{u}$ be this vector. Then:

(5)

Therefore we have that:

(6)

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