# Directional derivatives

### Directional derivatives

#### Lessons

Notes:

Suppose we have a vector $\vec{v} =\lt v_1, v_2\gt$. The unit vector will be:

$\vec{v} = \frac{1}{\sqrt{v^2_1 + v^2_2}} \lt v_1,v_2\gt$

Suppose we have a vector $\vec{v} =\lt v_1, v_2,v_3\gt$. The unit vector will be:

$\vec{v} = \frac{1}{ \sqrt{v_1^2 + v_2^2 + v_3^2} } $

When given an angle of a direction ($\theta$ ), we say that the unit vector (that points to the direction) is:

$\vec{u} = \lt \cos \theta, \sin \theta \gt$

Directional Derivatives of 2 Variable Functions
A Directional Derivative is the rate of change (of $x$ and $y$) of a function at a point $P=(x_0,y_0,z_0)$, at the direction of the unit vector

Suppose there is a 2-variable function $z=f(x,y)$. Then the directional derivative is:

$D_{\vec{u}}f(x,y) = f_x(x,y)a + f_y(x,y)b$

where the $\vec{u} = $ is the unit vector that points in the direction of change. Directional Derivatives of 3 Variable Functions
Suppose there is a 3-variable function $w=f(x,y,z)$. Then the directional derivative is:

$D_{\vec{u}}f(x,y,z) = f_x(x,y,z)a + f_y(x,y,z)b + f_z(x,y,z)c$

where the $\vec{u} = $ is the unit vector that points in the direction of change.
• Introduction
Directional Derivatives Overview:
a)
Things to Know Before Knowing Directional Derivatives
• Calculating unit vectors
• An example
• Angle to a unit vector
• An example

b)
Directional Derivatives of 2 Variable Functions
• The rate of change of $x$ and $y$
• $D_{\vec{u}}f(x,y) = f_x(x,y)a + f_y(x,y)b$
• An example

c)
Directional Derivatives of 3 Variable Functions
• The rate of change of $x$ and $y$
• $D_{\vec{u}}f(x,y,z) = f_x(x,y,z)a + f_y(x,y,z)b + f_z(x,y,z)c$
• An example

• 1.
Finding the Unit Vector & Angle of Direction
Find the unit vector of $\vec{v} =\lt 5, -2\gt.$

• 2.
Find the unit vector of $\vec{v} =\lt -1, 3, 5\gt.$

• 3.
Find the unit vector, given that the unit vector is in the direction of $\theta=\frac{\pi}{3}$.

• 4.
Finding the Directional Derivative of 2 Variable Functions
Find the direction derivative of $z=\sqrt{x^2+2y}$ at any given point, in the direction of $\vec{v} = \lt 1, 3\gt$

• 5.
Find the direction derivative of $z=xy \ln (\frac{x}{y})$ at any given point, where the direction of the unit vector is at $\theta=\frac{\pi}{6}$.

• 6.
Finding the Directional Derivative of 3 Variable Functions
Find the direction derivative of $f(x,y,z)=xy^3+yz^2$ in the direction of $\vec{v} =\lt 1, 2, 4\gt$.

• 7.
Find the direction derivative of $f(x,y,z)=\ln (x)e^{yz}$ in the direction of $\vec{v} =\lt -3, 1, 2\gt$ .