Directional derivatives

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:

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

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} = <a,b>

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} = <a,b,c>

is the unit vector that points in the direction of change.

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

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 $\vec{v} =\lt -3, 1, 2\gt$ .