Can't understand Leibniz notation?

Question

I just don't know what it's all about. OK, I know that (dy)/(dx) is just the derivative of y with respect to the x, but it gets more complicated when it gets to the chain rule and implicit differentiation. I don't know what does this notation represent. In fact, I don't even know what does d stand for. Can someone please explain it to me in the simplest possible terms (I tried Wikipedia but I still don't seem to get it)?

Answer 1

The letter "d" in Leibniz notation stands for "differential." So "dx" by itself means "differential of x." When you first learn about derivatives, you don't really need to know that, but it will come up a few weeks into the semeseter.

For very basic differentiation, you just memorize the fact that "dy / dx" is a whole symbol representing "the derivative of y with respect to x." But literally, it's the "differential of y divided by the differential of x." Still, with all of the formulas and shortcuts to finding derivatives, it's not necessary to think of it that way.

Later in the calculus class, you will have an opportuntiy to split up the differentials. You may have an equation that looks like this: dy = 3x^2 * dx. For more information, look up "differentials" in the index of your calculus book.

Why that notation? In algebra, you learn that the slope of a line (given two points) is (y2 - y1) / (x2 - x1). Another way of saying that is "change in y divided by change in x." In some books, you may see this written as (delta) y divided by (delta) x. (The Greek letter delta looks like a triange pointing upwards.) Traditionally in math, the delta symbol is used to represent changes in the value, particularly small changes.

The English equivalent of the Greek letter delta is "d." So by tradition, we use the Greek letter delta to represent actual changes in the variable when caluculating slope, and we use the "d" in front of the variable to represent theoretically small changes, the changes which occur when you take the limit in the definition of derivative.

As far as the Chain Rule is concerned, Leibniz notation is helpful because it _appears_ that we can use plain algebra on it. (There's more going on in the background, but the algebraic rules do apply.) So when you write [ dy / du ] * [ du / dx ], it appears that the du in the numertor can cancel with the du in the denominator, yielding dy / dx, which is in fact what it equals.

Leibniz notation is not absolutely required for implicit differentiation. You can get by just writing y' instead of dy/dx there. However, that's a part of related rates, and Leibniz notation is quite a bit more important in that topic.

Leibniz notation helps clarify what it is you're taking the derivative "with respect to." That's change in the function value _relative to_ change in a variable. That's very important when you have a function of more than one variable.

For example, y = x^2. It's easy enough to find y', that is, dy/dx. That's a rate; it shows how fast y changes for every step on the x-axis. (Derivative of y with respect to x.)

But suppose you're told that the x-coordinate changes at a rate of 2 ft/sec. That's also a rate, but it's distance with respect to time. We are now assuming that x and y are both functions of time instead of just y as a function of x.

So dx/dt is the rate of change of x relative to time. If x changes at 2 ft/sec, then dx/dt = 2.

"Find dy/dt." In this question, we no longer care how y moves relative to x. We care how y moves relative to t. So Leibniz notation makes it clear what variable is being measured (the numerator) and what variable it's being compared to (the denominator).

Answer 2

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RE:
Can't understand Leibniz notation?
I just don't know what it's all about. OK, I know that (dy)/(dx) is just the derivative of y with respect to the x, but it gets more complicated when it gets to the chain rule and implicit differentiation. I don't know what does this notation represent. In fact, I don't even know...

Answer 3

d stands for a really small change in the value of the variable. An infinitessimally small change.

When you find the slope of a line, you find change in y / change in x between two points. As the two points get closer together, the changes in x and y get smaller. For a line, though, the rate of change is the same for ANY two points.

For a curve, the rate of change changes. As the two points you use get closer together, you look at what the rate gets closer to. dy/dx is that limit. as the two points coincide the change in y and the change in x get really close to 0.