The Derivative

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The Derivative

• The Definition of the DerivativeSlope of the
  Tangent LineDerivative and DifferentiabilityDeri
  vatives GraphicallyDifferentiability and
  ContinuityDerivatives as Functions

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The Definition of the Derivative
Definition
Notations
Recall that by previous considerations, the
derivative of a function f at the point x0 is
the slope of the tangent line of the graph of f
at the point (x0,f(x0)). This is an important
geometric view to the derivative.
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Slope of the Tangent Line (1)
To compute the equation of the tangent line, at a
point (a,f(a)) of the graph of a function f
observe that
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Slope of the Tangent Line (2)
Example
Solution
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Slope of the Tangent Line (3)
Example
Solution (contd)
Conclusion
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Derivative and Differentiability (1)
Definition 1
Definition 2
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Derivative and Differentiability (2)
Theorem
Proof
?
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Derivative and Differentiability (3)
Theorem
Proof (contd)
?
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Differentiability Graphically
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Differentiability and Continuity
Theorem
Proof
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The Derivative as a Function
Definition
Notation
The derivative of a function f is a function
whose the domain of definition consists of all
the numbers x such that the function f has
its derivative at the point x.
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Graphs of Functions and their Derivatives
Problem
The picture on the right shows the graphs of two
functions f and g and the graphs of their
derivatives. The maximum of the shown values of
f is larger than that of g. Which is which?
Solution
To figure out which graphs are graphs of
functions, look at the points where the tangent
is horizontal. At such points the derivative
must vanish.
Which is function and which is derivative. Answer
is on the next slide.
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Graphs of Functions and their Derivatives
Df
Problem
f
The picture on the right shows the graphs of two
functions f and g and the graphs of their
derivatives. The maximum of the shown values of
f is larger than that of g. Which is which?
Solution
The red curve appears to intersect the x-axis
whenever the tangent of the blue curve is
horizontal. Hence the red curve is the graph of
the derivative of the function f. The blue
curve is the graph of f.
g
Dg
The green curve is everywhere decreasing. Hence
its tangents have negative slopes. The black
curve is the only curve that takes only negative
values. Hence the green curve is the graph of
the function g and the black curve that of its
derivative.