Download - Section 2.2
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Section 2.2
The Derivative as a Function
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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
h
xfhxfxf
h
)()(lim)(
0
NOTE: The function f ′ can be graphed and studied just like any other function.
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OTHER NOTATIONS FOR THE DERIVATIVE
Below are other notations for the derivative function, f ′(x). All of the these notation are used interchangeably.
)()(
)()(
xfDxDf
xfdx
d
dx
df
dx
dyyxf
x
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LEIBNIZ NOTATION
x
y
dx
dyx
0lim
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
axdx
dy
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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.
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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.
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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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