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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.

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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

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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.

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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.

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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.

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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

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