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Understanding the Derivative as a Function: Concepts and Notations

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revelin-nunan
8 diapositives · 235 Views · Nov 30, 2014

This section explores the derivative as a function, denoted as f′(x), providing a foundation for understanding calculus concepts. We discuss various notations including Leibniz notation (dy/dx) and its significance. Differentiability is defined, establishing that a function f is differentiable at a point a if f′(a) exists, and it is differentiable over an interval if it holds for every number within that interval. The relationship between differentiability and continuity is also examined, along with scenarios when a function may fail to be differentiable.

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Understanding the Derivative as a Function: Concepts and Notations
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  1. Section 2.2 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.