The Derivative Function (11/12/05) 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.

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.

Assignment Work on Hand-in Homework #3, which is due Friday at 4 pm. See if you can compute a formula for the derivative f ‘ (x ) for the functions f (x ) = x 4 and f (x ) = 1 / x 2