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Dot Notation / Newton's Notation — Definition, Formula & Examples

Dot notation, also called Newton's notation, writes derivatives by placing a dot above a variable. A single dot means the first derivative with respect to time, two dots mean the second derivative, and so on.

For a function x(t)x(t) of time, Newton's notation denotes the first time derivative as x˙\dot{x} and the second time derivative as x¨\ddot{x}, where x˙≡dxdt\dot{x} \equiv \frac{dx}{dt} and x¨≡d2xdt2\ddot{x} \equiv \frac{d^2x}{dt^2}. This notation is used almost exclusively when the independent variable is time.

Key Formula

x˙=dxdt,x¨=d2xdt2\dot{x} = \frac{dx}{dt}, \qquad \ddot{x} = \frac{d^2x}{dt^2}
Where:
  • xx = A function of time, such as position
  • tt = Time, the independent variable
  • x˙\dot{x} = First time derivative of x (e.g., velocity)
  • x¨\ddot{x} = Second time derivative of x (e.g., acceleration)

How It Works

You place one dot over a variable to indicate its first derivative with respect to time, and two dots for the second derivative. For example, if x(t)x(t) represents position, then x˙\dot{x} is velocity and x¨\ddot{x} is acceleration. The notation is compact but limited: it becomes impractical beyond the second derivative (three or more dots are hard to read and rarely used). It also does not easily indicate the variable you are differentiating with respect to, so it is assumed to be time unless stated otherwise.

Worked Example

Problem: A particle's position is given by x(t)=3t2+5tx(t) = 3t^2 + 5t. Find x˙\dot{x} and x¨\ddot{x}.
Step 1: Differentiate once with respect to time to get the velocity.
x˙=ddt(3t2+5t)=6t+5\dot{x} = \frac{d}{dt}(3t^2 + 5t) = 6t + 5
Step 2: Differentiate again to get the acceleration.
x¨=ddt(6t+5)=6\ddot{x} = \frac{d}{dt}(6t + 5) = 6
Answer: x˙=6t+5\dot{x} = 6t + 5 and x¨=6\ddot{x} = 6.

Why It Matters

Newton's dot notation is standard in physics and engineering whenever derivatives are taken with respect to time. You will encounter x˙\dot{x} and x¨\ddot{x} constantly in classical mechanics courses (e.g., Newton's second law written as F=mx¨F = m\ddot{x}) and in control theory, robotics, and differential equations involving dynamical systems.

Common Mistakes

Mistake: Using dot notation when the independent variable is not time.
Correction: Dot notation implicitly means differentiation with respect to time. For derivatives with respect to other variables (like xx or θ\theta), use Leibniz notation (dydx\frac{dy}{dx}) or prime notation (f′(x)f'(x)) instead.

Related Terms