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Section 3.2

Section 3.2. The Derivative as a Function. THE DERIVATIVE AS A FUNCTION. Given any number x for which the limit exists, we define a new function f ′( x ) , called the derivative of f , by. NOTE : The function f ′ can be graphed and studied just like any other function.

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Section 3.2

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  1. Section 3.2 The Derivative as a Function

  2. THE DERIVATIVE AS A FUNCTION Given any number x for which the limit exists, we define a new function f′(x), called the derivative of f, by NOTE: The function f′ can be graphed and studied just like any other function.

  3. OTHER NOTATIONS FOR THE DERIVATIVE Below are other notations for the derivative function, f′(x). All of the these notation are used interchangeably.

  4. LEIBNIZ NOTATION The dy/dx notation is called Leibniz notation. This notation comes from the increment notation; that is, If we want to indicate the value of dy/dx at the value a using Leibniz notation, we write

  5. DIFFERENTIABILITY Definition: A function f is differentiable at a if f ′(a) exists. The function f is differentiable on an open interval (a, b) [or (a, ∞) or (−∞, a) or (−∞, ∞) ] if it is differentiable at every number in the interval.

  6. DIFFERENTIABILITY AND CONTINUITY Theorem: If f is differentiable at a, then f is continuous at a. NOTE: The converse of this theorem is false; that is, there are functions that are continuous at a point but not differentiable at that point.

  7. HOW CAN A FUNCTION FAIL TO BE DIFFERENTIABLE? A function, f, can fail to be differentiable in three ways. (a) The graph can have a corner (or sharp point). (b) The graph can have a discontinuity. (c) The graph can have a vertical tangent.

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