2.9 - The Derivative As A Function

1. 2.9 - The Derivative As A Function

2. The First Derivative of f Interpretations • f ′(a) is the value of the first derivative of f at x = a. • f ′(x) is represents the instantaneous rate of change of f at x.

3. Definition A function f is differentiable at a if f ′(a) exists. It is differentiable on an open interval (a, b) [or (a, ∞) or (-∞, a) or (-∞, ∞)] if it is differentiable at every number in the interval.

4. Question If a function is discontinuous at a point, is it differentiable there? If a function is continuous at a point, is it always differentiable there? In other words, ff a function is continuous at a point, can we find the slope of the tangent line there? Consider the following functions. Determine the value of the derivative at the stated point.

5. Continuity and Differentiability Question: If f is differentiable at x = a within the domain of f, is it continuous at x = a? Theorem: If f is differentiable at a, then f is continuous at a. Caution: The converse to this theorem is not true. That is, if f is continuous at x = a, then it is not necessarily differentiable at x = a. Always think of f (x) = | x | at x = 0 if you get confused about this.

6. Notations Throughout history, different notations have been developed to represent the first derivative of f with respect to x. You should be familiar with them. is called Leibiniz Notation, where x and y (called differentials) represent infinitesimal changes in x and y, respectively.

7. Notations To evaluate a derivative using Leibiniz notation, we use the following notations.

8. Definition If limx→a | f ′ (a) | = ∞, then the graph of f may have a vertical tangent at x = a. If limx→a | f ′ (a) | = DNE, this can (but not always) indicate that the graph of f has a sharp point (called a cusp) at x = a. cusp point ∞ DNE