In multivariable calculus, the directional derivative is described in the following way:
While hiking in rugged terrain, you might think of your altitude at the point given by
longitude x and latitude y as defining a function $f (x, y)$. Although you won’t have a handy
formula for this function, you can say more about this function than you might expect. If
you face due east (in the direction of the positive x-axis), the slope of the terrain is given
by the partial derivative
${∂ f\over
∂x}
(x, y)$. Similarly, facing due north, the slope of the terrain is
given by
${∂ f\over
∂y}
(x, y)$. However, how would you compute the slope in some other direction,
say north-by-northwest? In this section, we develop the notion of directional derivative,
which will answer this question.
(Calculus, Early Transcendental Function, 4th ed. Smith and Minton)
Usually the diagram that is given is something like this:
all of which seem to make it very clear that this is a concept in multivariable calculus, such as in $R^3$.
I am wondering if such a concept exists in single variable calculus as well.
If I have a function such as $x^2$, we know how to find the rate of change at a point. But what about the rate of change at a point in the direction of a certain vector? Is there a reason why such a concept would not make sense?
I have in mind something like this:
Would it make sense to describe the rate of change at the green point in the direction of the red arrow?
Best Answer
For a single (real) variable function, points of the domain live on the real line, so in $\mathbb{R}$ and not in $\mathbb{R^2}$.
The only choice you still have, when approaching a point on the $x$-axis, is coming from the left- or the right-hand side. These one-sided derivatives (think about one-sided limits in general) are sometimes called the left and right derivative. You could see those as the only two possibilities in 1 dimension of the concept of "directions", corresponding to the "(unit) vectors" (in $\mathbb{R}$) $-1$ and $1$ respectively.