What is a directional derivative? I have encountered this in an online PDE course I'm following but I've never really been exposed to it. I've looked for the 'formal' definitions but I've never really understood any concept by looking at the formal, mathematical definition so can anyone elucidate this concept?
 A: The derivative is just the rate of change of a function of one variable.  Well, the directional derivative is the rate of change you get after converting a function of many variables into a function of one variable.  You do this by picking a "direction" and traveling along that direction.  As qaphla points out in a comment, what we call a "partial derivative" is exactly a directional derivative taken in a certain direction.  So the partial with respect to $x$ is the directional in the $x$ direction, and so on.
For example if I want to take a directional derivative of $f(x, y) = xy$ at $(a, b)$ in the direction $u = \langle 1, 2\rangle$ then my new function is
$$h(t) = f(a + t, b + 2t) = (a + t)(b + 2t) = ab + 2at + bt + 2t^2$$
and it's derivative is $h' = 2a + b + 4t$.  In $(a + t, b + 2t)$ the point $(a, b)$ corresponds to $t = 0$ so $\partial_uf(a, b) = h'(0) = 2a + b$.
On the other hand, if I want to find the partial with respect to $x$ then I choose the $x$ direction $u = \langle 1, 0\rangle$.  I get $h(t) = (a + t)b$, $h'(t) = b$, and so $\partial_xf(a, b) = b$.  Notice this is exactly what I get if I just follow the normal rules for differentiating $f$ and treat $x$ as a variable and treat $y$ as if it were a number (and hence had derivative $0$).
A: Suppose $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function. Then at a point $a\in\mathbb{R}$, the derivative $f'(a)$ tells you how quickly $f(x)$ increases (or decreases, if $f'(a)$ is negative) as $x$ increases, for $x$ near $a$. Now if we have a function $f:\mathbb{R}^n\to \mathbb{R}$, it doesn't make sense to talk about $x$ "increasing", but we can instead ask how $f(x)$ changes as $x$ moves in a prescribed direction $u$, for $x$ near a point $a\in \mathbb{R}^n$. This is what the directional derivative $\partial_uf(a)$ tells us.
