2.8 The Derivative As A Function

The Derivative if the limit exists. If f ’( a ) exists, we say f is differentiable at a. For y = f (x), we define the derivative of f at x, denoted f ’ (x), to be

Example f ( x ) = x 2 – 3x a)Find the derivative of f ( x ). b)Find an equation of the tangent line to f ( x ) at x = 2. c)If f ( x ) represents a position function for a moving vehicle, what does f’ ( x ) represent?

Interpretations of the Derivative If f ( x ) is a function, then f ’ ( x ) is ■ The formula for the slope of the tangent line to the graph of f ( x ). ■ the instantaneous rate of change of f ( x ) with respect to x. ■ the velocity function if f ( x ) is the position function of a moving object.

Theorem If f ( x ) is differentiable at a, then f ( x ) is continuous at a. Caution: If f is continuous at x = a, then it is not necessarily differentiable at x = a.

Nonexistence of the Derivative Some of the reasons why the derivative of a function may not exist at x = a are ■ The graph of f is not continuous at x = a. ■ The graph of f has a sharp corner at x = a. ■ The graph of f has a vertical tangent at x = a. If f is differentiable at a, its graph is “smooth” at a.

Notations is called Leibniz Notation Second Derivative  f ’( x ) is called the (first) derivative of f (x).  The derivative of f ’( x ) is called the second derivative of f (x), denoted by

Higher-Order Derivatives Third Derivative Fourth Derivative n th Derivative

Example f ( x ) = x 2 – 3x a)Find f’’ ( x ). b)Find f’’’ ( x ). c)Find the fourth derivative of f ( x ). d)If f ( x ) represents a position function for a moving vehicle, what does f’’ ( x ) and f’’’ ( x ) represent? e)Graph f ( x ), f’ ( x ), and f’’ ( x ).