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Delta x / Delta y: Definition, Examples

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What do Delta x / Delta y Mean?

The Greek letter Δ means “the change in” a variable. It’s usually used to mean a change in either the x-variable or y-variable:

  • x: a horizontal value on the x-axis.
  • Δx: a horizontal change (from point “a” to point “b” on the x-axis)
  • y: a vertical value on the y-axis.
  • Δy: a vertical change (from point “a” to point “b” on the y-axis)

Delta x / Delta y and The Slope

A linear (straight line) function has the equation
y = mx + b.
The “m” in the formula is the slope. In beginning algebra, you usually find the slope with the formula
slope = rise over run.
Where:

  • Rise = change in y-values (from point a to point b).
  • Run = change in x-values (from point a to point b).

delta x / delta y

“Rise over run” or Δy / Δx gives you the slope of a line.


Delta x / Delta y and Derivatives

The notation Δ x is used in calculus (especially physics) to mean tiny changes or very small increments. These tiny changes (close to zero) allow us to find the derivative, which is the slope at a point.

A similar notation is dy/dx, which means “derivative”. However, there’s a big difference between Δy / Δx, and dy/dx:

  • Δy / Δx, can be interpreted as meaning a quotient of small differences in y and x. The quotient is the answer to a division problem [1]. For example, in the problem 10 / 5, 2 is the quotient..
  • dy/dx means derivative and you can’t interpret it as a quotient of two numbers dy and dx.

The notation dy/dx is called Leibnitz notation. Another notation which means the same thing is f′ (prime notation), which avoids the confusion with a “quotient” [2].

Calculus suffers from a lack of consistent notation, partly due to so many different mathematicians working separately in the early years of its development. There are other various meanings for delta (both upper- and lower-case) including “discriminant”. See: Delta meaning.


References

[1] Open University. 3.3.3 Multiplication and Division Math Terminology.
[2] Stony Brook Calculus Materials: Calculus I Handouts.

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Stephanie Glen. "Delta x / Delta y: Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/delta-x-delta-y-definition-examples/
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