The phrase "instantaneous rate of change of f with respect to x at a" is a mouthful, because there are a lot of things we need to specify in order to be precise. This is why scientists sound like they're speaking a different language sometimes. They are. We need to specify
- that we're finding the instantaneous rate of change (instead of the average rate of change)
- what the independent variable is (often, but not always, x)
- what the dependent variable is (often y or a function f(x))
- the particular place we'd like to find the instantaneous rate of change (the value a of x)
Since we need to make all these things clear, it's a little hard to say this awful mouthful with fewer words. This is where mathematical notation comes in handy. Instead of saying "instantaneous rate of change of f with respect to x at a" we can say f'(a) or
Mathematic notation is a language all its own, and sometimes several languages. A translation guide follows.
Prime Notation
We've mostly been using the notation f ' (a) to denote "the derivative of f at a." If f is a function of x, this means the same thing as "the instantaneous rate of change of f with respect to x at a."
When using this kind of notation, we talk about "the instantaneous rate of change of f at a." We don't bother to mention x because the independent variable is clear from context. When the variables aren't clear from context, we need to use different notation.
The notation f ' (a) is known as Lagrange notation. It's named for Joseph-Louis Lagrange, who, besides inventing a tidy way of writing things, lived through the French Revolution and the Reign of Terror.
Fraction Notation
Another way of writing instantaneous rates of change is with fractions, better known as Leibniz notation. Leibniz was another mathematician. He and Isaac Newton were total frenemies.. He respected him, but disliked him for beating him to the punch line that is calculus. At least his notation is still a crowd pleaser.
Leibniz notation looks like this:
This notation is nice because it looks like we're talking about a slope (which we are, since instantaneous rate of change is the same thing as slope at a point). The "rise" variable, y, is on the top; the "run" variable, x, is on the bottom:
We can think of the expression
as
We've already said this notation is nice because it reminds us we're talking about slopes. It also allows us to specify variables more carefully. When we start talking about multiple variables that relate to each other, we might want to talk about quantities such as the rate of change of x with respect to y. It's not immediately clear how we would do this with Lagrange notation, but with Leibniz notation, it's do-able. The instantaneous rate of change of x with respect to y when y = b is
We could also write the instantaneous rate of change of x with respect to y when x = a:
We can also translate notation back into words. We find the instantaneous rate of change of (the variable on the top) with respect to (the variable on the bottom). The expression
means "the instantaneous rate of change of K with respect to R when R = 2."
Finally, here's one other nice thing we find from Leibniz notation: it's possible to see what the units of the instantaneous rate of change (also known as the derivative) are. The units of the quantity
are the units of y divided by the units of x:
So if
is measured in miles per hour then
is measured in hours per mile.
We take the units of the variable on top, and divide by the units of the variable on the bottom. The bit at the lower right that says x = a doesn't affect the units.
Sample Problem
If y is in miles and x is in hours, then
is measured in miles per hour and
is measured in hours per mile.
We take the units of the variable on top, and divide by the units of the variable on the bottom. The bit at the lower right that says x = a doesn't affect the units.