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lecture12 - Lecture 12 The Derivative of a Function(Chapter...

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Lecture 12: The Derivative of a Function (Chapter 2, Sections 11 and 12) Recall: the slope of the tangent line to a function f ( x ) at x = a is given by This limit also gives us the instantaneous rate of change of f at x = a . Def. The derivative of a function f at x = a is deflned to be if the limit exists. If h = x ¡ a , then f 0 ( a ) =

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2 The equation of the tangent line to y = f ( x ) at x = a : ex. Find the equation of the tangent line to f ( x ) = 2 x ¡ x 2 at x = 2. 6 - ?
3 The derivative as a function Def. Given y = f ( x ), f 0 ( x ) = The derivative is itself a function of f . Its domain: Other notations for the derivative: Process of flnding the derivative is called

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4 ex. Find the function f 0 ( x ) for f ( x ) = x 2 ¡ x . What is its domain? Def. A function f is difierentiable at x = a if f 0 ( a ) exists. It is difierentiable on an open interval if it is difierentiable at each number in the interval. ex. Find each interval for which f ( x ) = x 2 ¡ x is difierentiable.
5 ex. Let f ( x ) = ( 2 ¡ x x 1 x 2 x > 1 .

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