Definition of Derivative

Langara College - Department of Mathematics and Statistics

Internet Resources for the Calculus Student - Topics in Calculus

Definition of the Derivative

The derivative of a function is another function. Namely the one whose value at a point is equal to the rate of change of the given function (so, for a moving particle, the derivative of its position as a function of time is its velocity as a function of time). Here, by "rate of change" we mean the "instantaneous rate of change" at a particular point rather than the average rate over an interval, and there are various on-line illustrations of this distinction, and of how the instantaneous rate can be defined as a limit of average rates. Geometrically, in terms of the graph, the instantaneous rate of change corresponds to the slope of the tangent line. Several on-line animations and applets are available to illustrate how this varies as we move along the curve, and also to show how it can be found as a limit of secant line slopes.

The precise formal definition of the derivative avoids the term "instantaneous rate" by working directly in terms of the limit of average rates (which are themselves just ratios of differences).

Definition

For any function, f , its derivative is the function, f ', whose value at any point x is given by

f(x+h) - f(x)

f ' (x) = lim

h -> 0 h

(with domain D(f ')={all x for which the above limit exists}).

In the above definition the symbol h refers to a quantity that is being taken to a particular limiting value (in this case 0), and since h is not a value on which the limit depends, it is often called a "dummy variable" and could be replaced by any other symbol without affecting the result. A popular choice is to use instead the expression 'delta x' to emphasize that it referrs to a difference of x values.

Another equivalent version of the definition is to work in terms of the displaced point x+h (let's call it z) rather than the displacement h, and to consider the limit as z->x of the ratios (f(z)-f(x))/(z-x)

In the above limit, the values of h can be either positive or negative, and for f to be "differentiable" at x it is necessary for this two sided limit to exist. But we could also consider the "one-sided derivatives" that one might define by looking at just positive or just negative values of h. This concept is illustrated at derivOneSide@ies

The name f ' that we have used for the derivative of f is similar to that used by Newton (except that he used a dot directly over the f rather than a 'prime' or apostrophe).

Another notation that is often used is due to Leibnitz, the other main inventor of Calculus.
Instead of f', Leibnitz wrote df/dx (which is not intended to make sense of df and dx separately, but just to remind us of the definition as a limit of ratios). One problem with Leibnitz' notation is that it depends on having a name for the input variable (x), and that is not really a property of the function f itself, as any real value in its domain can be used as input. This can sometimes be a source of confusion, especially when dealing with composite functions and the chain rule, so watch out!

Some modern texts use the notation Df rather than f ' . This emphasizes the idea of differentiation as an operation on the function f, and allows us to refer to that operation itself with the symbol D.
When we want to emphasize differentiation as an operation, in terms of Leibnitz' notation we often write df/dx as (d/dx)f, so the differentiation operator D above can be written as d/dx.

If you have come across any good web-based discussions of the definition of the derivative,
please do let us know and we will add them here.

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Derivatives

Internet Resources for the Calculus Student - Topics in Calculus

Definition of the Derivative

Definition

Derivatives

Intro to Concept and Definition

Tangent Slopes

Secant Approximation to Tangent Lines

Avg&Inst Rates

Differentiability

Interpretation

Some Basic Rules

Exp&Log

Trig

Chain Rule

Mean Value Theorem