Differential Notation

\dot{f}(t) = \lim_{\Delta t \to 0} \frac{f(t+\Delta t) - f(t)}{\Delta t}

This "dot" notation is one of two common ways of representing the derivative.

Calculus was simultaneously invented by two people: Gottfried Wilhelm Liebniz and Isaac Newton.

And each came up with his own notation for representing derivatives. The Wikipedia article on Notation for Differentiation does a good job of explaining them thoroughly but I will summarize here.

In both notations, \frac{d}{dt} is an instruction to take the derivative. It means "Take the derivative with respect to t of whatever function shows up to the right."

When you see something like this:

\frac{df}{dt}

You should think "the derivative of some function f with respect to t"

1. Liebniz Notation (prime)

If some variable y is a function of x we can write:

y=f(x)

The derivative of y with respect to x is given by:

\frac{dy}{dx} = f '(x)

and this could be spoken as "dee y dee x equals f prime of x"

We will not be using this notation in this Nanodegree.

2. Newton's Notation (dot)

Newton invented Calculus as a tool to help him understand motion. As a result, he was usually thinking of derivatives with respect to time (not some abstract x variable).

Likewise, his functions weren't abstract f(x)'s and g(f(x))'s. The functions he was interested in actually meant something about the physical world! He wanted to describe:

position x(t)

velocity v(t)

and acceleration a(t)

And he wanted to capture the relationships between these quantities compactly. So for Newton: differentiation with respect to time is indicated by placing a dot over the variable.

So, for example:

v(t) = \frac{d}{dt}x(t) = \dot{x}(t)

or for second derivatives:

a(t) = \frac{d}{dt} v(t) = \frac{d}{dt} \dot{x}(t) = \ddot{x}(t)

A second derivative can also be represented as follows:

a(t) = \frac{d^2}{dt^2} x(t) = \frac{d^2x}{dt^2}

Next Concept

Differential Notation

0. What Newton and Liebniz share (d/dt)

1. Liebniz Notation (prime)

2. Newton's Notation (dot)