Calculus III:

Partial derivatives
Directional derivatives

1. The unit vector directional
of the change of a function

Directional derivatives are derivatives along a unit vector.

The two partial derivatives f_x and f_y represent respectively the rate of change of f as we vary x and holding y fixed; and as we vary y and holding x fixed .

Now, we need to know what is the rate of change of the function f if we change both x and y simultaneously.

So, let's suppose that we want te define the rate of change of f at a particular point, say (xo, yo).
Let’s also suppose that x changes by α and y changes by β; that is
xo becomes x = x0 + αh and
yo becomes y = yo + βh.

The parameter h is the measure of the change.

For example if the unit vector = 〈1/√10, 3√10〉, the point (xo, yo) becomes the point (xo + h/√10, yo + 3h/√10).

α and β are the coordinates of the unit vector at the point (xo,yo). They are the units of measure of the change along x-axis and y-axis.

The direction of the vector is the direction of the variation of change.

Note that if we want the direction of change as an angle θ say θ = π/4, we express the unit vector in termes of this angle, that is

= 〈cos θ, sin θ 〉

2. Definitions

The derivative of a function of a single variable is the rate of change defined by :

Df(x) = lim (f(x + h) - f(x))/h
h → 0

That can be written as a directional derivative

D f(x) = lim (f(x + αh) - f(x))/h
h → 0

Similarly,

The derivative of a function of two variables is its rate of change with x and y, defined by :

Df(x,y) = lim (f(x + αh, y + βh) - f(x,y))/h
h → 0

Written as

D f(x,y) = lim (f(x + αh, y + βh) - f(x,y))/h
h → 0

The definition of the directional derivative is :

The rate of change of f(x,y) in the direction of the unit vector = 〈α, β〉 is called the directional derivative, denoted by D f(x,y) and defined by

D f(x,y) = lim (f(x + αh, y + βh) - f(x,y))/h
h → 0

3. Explicite form of the directional derivative

We want to write down a formula much simpler to use than the limit definition presented above.

Let’s define a new function of a single variable of z g(z) = f(xo + αz, yo + βz)

By the definition of the derivative for functions of a single variable we have:

g'(z) = lim (g(z + h) - g(z))/h
h → 0

g'(0) = lim (g(h) - g(0))/h
h → 0

We have

g(h) = f(xo + αh, yo + βh),
g(0) = f(xo, yo)

Therefore

g'(0) = lim (f(xo + αh, yo + βh))/h = D f(xo,yo)
h → 0

So

g(z) = f(x,y) = f(xo + αz, yo + βz),
where x = xo + αz , and y = yo + βz

Using the chain rule, we have

g'(z) = dg(z)/dz = (∂f/∂x)(dx/dz) + (∂f/∂y)(dy/dz) =
(∂f/∂x)(α) + (∂f/∂y)(β) = αf_x + βf_y

g'(z) = αf_x(x,y) + βf_y(x,y)

So,taking z = 0, we obtain

g'(0) = Df(xo,yo) = αf_x(xo,yo) + βf_y(xo,yo)

Hence, for any x and any y, we have the following formula:

g'(z) = Df(x,y) = αf_x(x,y) + βf_y(x,y)

Df(x,y) = αf_x + βf_y



Working with functions with more than two variables, the formula to use is similar to found for function with two variables.

For instance, the directional derivative of f(x,y,z) in the direction of the unit vector (α β γ) is given by,

Df(x,y,z) = αf_x + βf_y + γf_z

Example

Find the directional derivative D f(0,1), where f(x,y) = 2 xy , and is the unit vector in the direction θ = π/4.

= 〈cos (π/4), sin (π/4) 〉 = 〈√2/2, √2/2 〉

f_x = 2y, and f_y = 2x. Then

D f(x,y) = αf_x + βf_y = (√2/2) 2y + (√2/2) 2x = √2(x + y)

D f(x,y) = √2(x + y) . Therefore

D f(0,1) = √2(x + y) = √2.

D f(0,1) = √2.

4. Other formula for the directional derivative

Df(x,y,z) = αf_x + βf_y + γf_z

This expression of the directional derivative can be written as the dot product of ∇f and :

Df(x,y,z) = ∇f .

is the unit vector that gives the direction of change = 〈α,β,,γ〉.

∇f is the gradient vector of f(x,y,z) = 〈f_x,f_y,f_z〉

5. Naximum rate of change of a function

Df(x,y,z) = ∇f .

Takinng the magnitude of this dot poduct and because the directional vector is a unit vector, we have:

||Df(x,y,z) = ∇f . || =
||∇f|| |||| cos φ = ||∇f|| cos φ .

φ is the angle between ∇f and .

||Df(x,y,z) = ||∇f|| cos φ

The largest possible value of φ is 0. This is the direction that we need to move in order to achieve that maximum rate of change. will point in the same direction as the gradient ∇f.

The maximum rate of change of the function f(x,y,z), that is the maximum value of Df(x,y,z) is given by ||∇f|| and occur in the direction of ∇f.