Calculus How To

Function of One Variable & Ordinary Derivative

Types of Functions >

Contents:
Function of One Variable
Ordinary Derivative

What is a Function of One Variable?

A function of one variable calculates one output value for one input value. For example, f(x) = 2x has one variable (x) and equals 4 for an input of 2:

  • f(x) = 2x
  • f(2) = 2(2) = 4

Function of One Variable Characteristics

A function of one variable has three defining characteristics:

  1. It’s a function (i.e. one input results in exactly one output),
  2. It has a single variable, like “x” or “t”.
  3. The functions deal with real numbers (as opposed to complex/imaginary numbers like 4i ).

Examples:

  • f(x) = x2 + 5,
  • f(x) = 2x + 5,
  • f(t) = t3 + 9t,
  • f(t) = t3 + t2t + 10.

Notice how all the examples have just one variable in the expression: either x or t. While x and t are the more common variables, you might see others. For example, in a trigonometric function like the sine function, you might see an angle (θ) instead, e.g. f(x) = sin(θ).

If you’re just starting out with calculus, you’re working with functions of one variable, even if it isn’t explicitly stated in your textbooks or by your professor. A function of one variable is the “common” function you’ll deal with, right up until you hit multivariable calculus and complex analysis.

Function of One Variable: Types

A function of one variable fits into a subset of different types. There are many dozens of different one-variable functions, but some of the more common ones you’ll come across include:

Constant functions: have the form f(x) = c, where “c” is a constant, like 5, 44, or 99.99.

constant function of one variable

The constant function y(x) = 1 is an example of a function of one variable.


Linear Functions: functions that produce a straight line graph.

linear function of one variable

A linear function graphs a straight line.


Ordinary Derivative vs. Partial

An ordinary derivative is a derivative that’s a function of one variable, like F(x) = x2. The purpose is to examine the variation of the function with respect to one variable (x, in this example).

ordinary derivative of a constant function

Ordinary derivative of a constant function.

One of the more popular subtypes of derivatives is the partial derivative, so the “ordinary derivative” is sometimes defined in terms of a partial derivative. This isn’t strictly true (it can be a comparison with any derivative, as demonstrated below). But, for clarity, here’s the main difference between the two:

A partial derivative has more than one variable, such as F(x, y) = ax2 + by2. With a partial derivative, you hold one variable constant, in order to examine the variation of the function with respect to the other. In this example, you could hold x constant while examining y.

The term ordinary derivative is an informal term usually used to denote a “usual” or “regular” derivative, to distinguish it from other types of derivatives. For example:

  • Covariant derivatives (which operate in relative complex geometrical dimensions),
  • Directional derivatives. For example “This also makes our definition of the directional derivative coincide with the ordinary derivative in the one dimensional case” (Engleberg, 2018).
  • Partial derivatives,
  • The q-derivative (also called a Jackson derivative), a q-analog of the ordinary derivative, used in combinatorics and quantum calculus.
  • Riemann derivatives—used in fractional calculus, or
  • The somewhat obscure Peano derivative. For example, from Butzer et al. (1971): “Show that if the rth ordinary derivative…exists, so does the rth Peano derivative….

Many more other types of derivatives exist. So the term “ordinary” is inserted by authors just to make it clear which derivative is which.

References

Ash, J. A Characterization of the Peano Derivative. Transactions of the American Mathematical Society. Volume 149, June, 1970
Butzer, P. et al. (1971). Fourier Analysis and Approximation.
Engleberg, S. (2018). Random Signals and Noise: A Mathematical Introduction.
Larson, R. & Edwards, B. (2009). Calculus. Cengage Learning.
Nave, C. (2016). The Derivative. Retrieved December 30, 2019 from: http://hyperphysics.phy-astr.gsu.edu/hbase/deriv.html
Oldham, K. et al. (2008). An Atlas of Functions: with Equator, the Atlas Function Calculator 2nd Edition. Springer.

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