Directional Derivatives Examples 5

# 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)
\begin{align} \quad D_{\vec{u}} \: f(x, y) = \frac{\partial z}{\partial x} a + \frac{\partial z}{\partial y} b \end{align}

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)
\begin{align} \quad D_{\vec{u}} \: f(x, y, z) = \frac{\partial w}{\partial x} a + \frac{\partial w}{\partial y} b + \frac{\partial w}{\partial z} c \end{align}

## 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)
\begin{align} \quad \vec{u} = \frac{(4, 2)}{\| (4, 2) \|} = \frac{(4, 2)}{\sqrt{20}} = \frac{(4, 2)}{2 \sqrt{5}} = \frac{\left (2, 1 \right )}{\sqrt{5}} = \left ( \frac{2}{\sqrt{5}}, \frac{1}{\sqrt{5}}\right ) \end{align}

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

(4)
\begin{align} \quad D_{\vec{u}} \: f(x, y) = \frac{2}{\sqrt{5}} \frac{\partial}{\partial x} (y^3\ln x) + \frac{1}{\sqrt{5}} \frac{\partial}{\partial y} (y^3\ln x) \\ \quad D_{\vec{u}} \: f(x, y) = \frac{2}{\sqrt{5}} \frac{y^3}{x} + \frac{1}{\sqrt{5}} 3y^2 \ln x \end{align}

## 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)
\begin{align} \quad \vec{u} = \frac{(3, 1)}{\| (3, 1) \|} = \frac{(3, 1)}{\sqrt{10}} = \left ( \frac{3}{\sqrt{10}}, \frac{1}{\sqrt{10}} \right ) \end{align}

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

(6)
\begin{align} \quad D_{\vec{u}} \: f(x, y) = \frac{3}{\sqrt{10}} \frac{\partial}{\partial x} (x^2 + y^2) + \frac{1}{\sqrt{10}} \frac{\partial}{\partial y} (x^2 + y^2) \\ \quad D_{\vec{u}} \: f(x, y) = \frac{3}{\sqrt{10}} 2x + \frac{1}{\sqrt{10}} 2y \end{align}