Objectives: Now that we have defined the derivative of a function at a point, in this tutorial, we define a function which is the derivative at all points of an interval. We use the definition of a derivative to find the derivative of some functions. We also define the concepts of right-hand and left-hand derivatives and apply these concepts to piecewise defined functions. Various notations for derivative are shown. The relationship between differentiability and continuity are explored.

After working through these materials, the student should be able

• to use the definition of derivative to find the derivative of several functions; and
• to understand the relationship between differentiability and continuity.

Modules:

 Definition. Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit provided this limit exists. If this limit exists for each x in an open interval I, then we say that f is differentiable on I.

• Examples.

 Definition. Let y = f(x) be a function and let a be in the domain of f. The right-hand derivative of f at x = a is the limit and the left-hand derivative of f at x = a is the limit The function f is differentiable on the interval I if when I has a right-hand endpoint a, then the left-hand derivative of f exists at x = a, when I has a left-hand endpoint b, then the right-hand derivative of f exists at x = b, and f is differentiable at all other points of I.

• As a consequence of Limit Theorem C, we have the following.

 Corollary. A function f is differentiable at x = a if and only if f has both a right-hand derivative and a left-hand derivative at x = a and both of these derivatives are equal.

• Examples.

• Notation

We have used the notation f' to denote the derivative of the function f. There are also many other ways to denote the derivative.

• Df and Dxf are used by some authors to denote the derivative of the function f.
• If we consider y = f(x), then y' denotes the derivative of the function f.
• If we let Dx = h and Dy = f(x + h) - f(x) then denotes the derivative of the function f.

• A variation of the previous is • The following theorem relates the two properties of differentiability and continuity.

 Theorem. If the function f is differentiable on the interval I. then f is continuous on I.

• Discussion [Using HotEqn] [Using IBM Techexplorer] [Using IBM Pro. Techexplorer]

• Example. We have seen (Example 2 after Theorem C) that the function f(x) = |x| is continuous at x = 0 but is not differentiable at x = 0. Hence, the converse of the Theorem is not true.