# Directional Derivatives Examples 5

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

**Determine the directional derivative of $f(x, y) = y^3\ln x$ in the direction of the vector $(4, 2)$.**

The vector $(4, 2)$ is not a unit vector. The unit vector that points in the direction of $(4, 2)$ is given by:

(3)

Therefore we have that the directional derivative of $f$ in the direction of $\vec{u}$ is given by:

(4)

## Example 2

**Find the directional derivative of $f(x, y) = x^2 + y^2$ in the direction of $(3, 1)$.**

Once again we have that $(3, 1)$ is not a unit vector. The unit vector in the direction of $(3, 1)$ is given by:

(5)

Therefore we have that the directional derivative in the direction of $(3, 1)$ is:

(6)

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