Directional Derivatives

How do we find the rate of change of $f$ if we allow both $x$ and $y$ to change simultaneously?

Defining Direction of Change

The problem here is that there are many ways to allow both $x$ and $y$ to change; for example, one could be changing faster than the other, or one could be increasing while the decreasing. Thus, we need to find a good way to define the changing of $x$ and $y$.

Let’s say we want to the rate of change of $f$ at a point $(x_0,y_0)$, where $x$ and $y$ are both increasing but $x$ is changing twice as fast as $y$; as $y$ increases by one unit of measure, $x$ increases by two.

If we have a particle sitting at point $(x_0,y_0)$ and the particle will move in the direction given by the changing $x$ and $y$; thus, the particle will direction of movement can be defined as a vector:

$$\vec{v}=⟨2,1⟩$$

However, this is not specific enough as there are many dependent vectors that point in the same direction. Thus, we should insist that the unit vector should be used:

$$\vec{v}=⟨\frac{2}{\sqrt{5}},\frac{1}{\sqrt{5}}⟩$$

Alternatively, we can give the direction of changing $x$ and $y$ as an angle. In this case, the unit vector can be defined in as:

$$\vec{v}=⟨\cos \theta ,\sin \theta ⟩$$

Finding Rate of Change

Definition: Directional Derivative

The rate of change of $f(x,y)$ can in the direction of unit vector $\vec{u}=⟨a,b⟩$ is called the directional derivative and is:

$$D_{\vec{u}}(x,y)=\underset{h\to 0}{\lim }\frac{f(x+ah,y+bh)-f(x,y)}{h}$$

This is very similar to partial derivatives, but can be difficult to compute in practice, so it’s useful to derive an equivalent formula for taking directional derivatives, where $\vec{u}=⟨a,b⟩$ is the direction:

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

This also expands to more than 2 variables. For example, the directional derivative of $f(x,y,z)$ in the direction of unit vector $\vec{u}=⟨a,b,c⟩$ is given by:

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

Derivation of above formula

Let’s define a new function of a single variable:

$$g(z)=f(x_0+az,y_0+bz),$$

where $x_0,y_0,a,b$ are fixed numbers; $z$ is the only variable. Using the traditional definition for a single-variable derivative, we have:

$$g^{\prime }(z)=\underset{h\to 0}{\lim }\frac{g(z+h)-g(z)}{h},$$

and the derivative at $z=0$ given by,

$$g^{\prime }(0)=\underset{h\to 0}{\lim }\frac{g(z+0)-g(z)}{h}$$

Substituting in $g(z)$, we have:

$$\begin{array}{rl}{g}^{\prime }(0)& =\underset{h\to 0}{\lim }\frac{g(h)-g(0)}{h}\\ & =\underset{h\to 0}{\lim }\frac{f(x_0+ah,y_0+bh)-f(x_0,y_0)}{h}\\ & =D_{\vec{u}}f(x_0,y_0)\end{array}$$

Now with $g(z)$ instead:

$$g(z)=f(x,y)\text{where}x=x_0+az\text{and}y=y_0+bz$$

Using the chain rule, we can expand this to:

$$\begin{array}{rl}{g}^{\prime }(z)& =\frac{dg}{dz}=\frac{\partial f}{\partial x}\frac{dx}{dz}+\frac{\partial f}{\partial y}\frac{dy}{dz}\\ & =f_x(x,y)a+f_y(x,y)b\end{array}$$

If we take $z=0,x=x_0,y=y_0$:

$$g^{\prime }(0)=f_x(x_0,y_0)a+f_y(x_0,y_0)b$$

Which gives us:

$$D_{\vec{u}}f(x_0,y_0)=g^{\prime }(0)=f_x(x_0,y_0)a+f_y(x_0,y_0)b$$

If we go back to allowing $x$ and $y$ to be any number we get the above formulas for computing directional derivatives

There is also another form that is a little nicer and more compact; it’s also much more general and encompasses the above forms. We can rewrite the above formula as:

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

As such, we can write the directional derivative as a dot product and notice that the second vector is nothing more than the unit vector $\vec{u}$ that gives the direction of change.

We can also write this in terms of gradient vectors:

The above definitions are for functions of 3 variables, but of course this extends to any number of variables

Link to original

As such, we can say that the directional derivative can be given by

$$D_{\vec{u}}f=\nabla f\cdot \vec{u}$$

This is more general, as we don’t need to show the variable and use this formula for any number of variables. Alternatively, we can use the notation:

$$D_{\vec{u}}f(\vec{x})=\nabla f\cdot \vec{u}$$

where $\vec{x}=⟨x,y,z⟩$.