Clicker Question 1 What is an equation of the tangent line to the curve f (x ) = x 2 at the point (1, 1)? A. y = 2x B. y = 2 C. y = 2x 2 D. y = 2x + 1 E. y = 2x – 1

The Derivative Function (2/13/09) Since we know how to compute (or at least estimate) the derivative (i.e., the instantaneous rate of change) of a given function f at any fixed point, we can then view the derivative itself as a function . We denote this function by f ' (we say “f prime”). Given an input x , it outputs the rate of change at x .

The Derivative Function Definition: The derivative function can be : Graphed, given a graph of f Estimated, given numerical info about f Computed algebraically, given a formula for f

Derivatives of very simple functions What is the derivative of any constant function f (x) = c at any point x ? What is the derivative of any linear function f (x) = m x + b at any point x ? That is, again, you don’t need calculus to understand the rate of change of linear functions.

Derivatives of Some Algebraic Functions Use the definition of the derivative to find the derivative of the function f (x) = x 3 Use the definition to find the derivative of f (x) = 1 /x . Use the definition to find the derivative of f (x) = x .

Clicker Question 2 Using what we just learned, what is the instantaneous rate of change of the function f (x ) = x 3 at the point (2, 8)? A. 3x 2 B. 8 C. 12 D. 12x E. 12x 2

Differentiability A function f is said to be differentiable at x = a if f '(a ) exists. How can f not be differentiable at a ?? If the graph of f breaks apart at a (we say f is not continuous at a ). If the graph has a sharp bend at a . If the graph becomes vertical at a .

Clicker Question 3 The function f (x ) = |x | is not differentiable A. at x = -1 B. at x = 0 C. at x = 1 D. at all three of these points E. everywhere

Assignment We will have Lab #3 on Monday on rates of change of functions. For Wednesday, please read Section 2.8 and do Exercises 1, 3, 5, 9, 14, 16, 19, 21, 25, 27, 29, 33a, 35, and 51.