# Higher Order Directional Derivatives

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

(1)

We saw that we could compute directional derivatives of $f$ with the following formula:

(2)

We will now look at computing higher order directional derivatives. The process is much the same as computing higher order partial derivatives. Suppose that we have a function $z = f(x, y)$ and a unit vector $\vec{u} = (a, b)$. Then the first directional derivative of $f$ in the direction of $(a, b)$ is $D_{\vec{u}} \: f(x, y) = a \frac{\partial z}{\partial x} + b \frac{\partial z}{\partial y}$ as noted above. The directional derivative in the direction of $\vec{u}$ will define a function, say:

(3)

If we want to take the second directional derivative of $f$ in the direction of $(a, b)$, then we have that:

(4)

If the second partial derivatives of $f$ are continuous, then by Clairaut’s theorem we have that:

(5)

A formula for the second directional derivative of a three variable real-valued function $w = f(x, y, z)$ can be obtained in a similar manner.

Of course, we can take successively higher order directional derivatives if we so choose. It’s not practical to remember the formulas for computing higher order direction derivatives of a function of several variables though.

Let’s look at an example of finding a higher order directional derivative.

## Example 1

**Find the second order and third order directional derivative of the function $f(x, y) = 2xy^2$ in the direction of $(1, 2)$.**

The vector $(1, 2)$ is not a unit vector. We have that the unit vector in the direction of $(1, 2)$ is given by:

(6)

Therefore the first directional derivative in the direction of $(1, 2)$ is given by:

(7)

We will now take the second directional derivative in the direction of $(1, 2)$:

(8)

Lastly, we find the third order directional derivative as follows:

(9)