Directional derivatives of a function of 2 variables

 Goal: Determine the directional derivative of a function of two  variables

Considerations:

In a function of two variables, we have so far considered partial derivatives both with respect to x and y. In these partial derivatives, only one variable is changing. In partial derivative with respect to x, this variable is changing while y is constant. In partial derivative with respect to y only y is changing while x is constant.

In directional derivatives, both variables x and y are changing. The changing of  these variables provide a direction. This direction is represented by a vector expressed in function of an angle.

We consider the graph of a surface represented by the function z = f(x, y). We consider a point (a,b) that belongs to the domain of f.  The direction of travel starts from that point and is measured according to an angle θ, directed counterclockwise in the xy plane starting at zero from the positive direction of the x-axis. The distance traveled is h and the direction is given by the vector u = cos(θ)i + sin (θ)j, The z coordinate of the second point on the graph  is given by z  =  f( a + hcosθ, b + hsinθ,).

We start from a point (a, b, f(a,b)) of the surface and arrive at a second point of which the z coordinate is  z  =  f( a + hcosθ, b + hsinθ,). The slope of the secant line joining these two points is found by dividing the difference of the z coordnates by the difference traveled, which is h. We have:

The directional directive of the function f in the direction u is equal to the slope of the tangnt line at the given point. The slope is found by taking the limit of the above expression when h approaches zero,

Definition

Suppose z = f(x.y) is a function of two variables with a domain D. Let (a, b) ϵ D and define u = cos(θ)i + sin (θ)j. Then the directional derivative in the direction of u is given by:

provided that the limit exists.

Example

Let θ = arccos(3/5). Find the directional derivative Dᵤf(x,y) of the function f(x, y) = x² - xy + 3y² in the direction of u = cos(θ)i + sin (θ)j,. What is Df(-1, 2)?

Solution

According to the definition, Dᵤf(x,y) is given by:

Dᵤf(x,y) = lim f(x + hcosθ, y + hsinθ) - f(x,y)/h when h approaches zero. Let's start by calculating 

f(x + hcosθ, y + hsinθ):

Since cos θ = 3/5, sin θ is given by:

Let's substitute sin θ and cos θ :

Let's substitute f(x + hcosθ, y + hsinθ) and f(x, y) in the expression: Dᵤf(x,y) = lim f(x + hcosθ, y + hsinθ) - f(x,y)/h when h approaches zero: We have:

Dᵤf(x,y) 

To fnd Dᵤf(-1, 2)? let's substitute x by -1 and y by 2 in the above expression:

See the following figure:

In the figure above we can see that the plane is tangent to the surface at the point (-1,2,15).