Section 2'8 The Derivative as a Function

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Section 2.8The Derivative as a Function

• Goals
• View the derivative f?(x) as a function of x
• Study graphs of f?(x) and f(x) together
• Study differentiability and continuity
• Introduce higher-order derivatives

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Introduction

• So far we have considered the derivative of a
  function f at a fixed number a
• Now we change our point of view and let the
  number a vary

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Introduction (contd)

• Thus f?(x) becomes its own, new, function of
  x , called the derivative of f .
• This name reflects the fact that f? has been
  derived from f .
• Note that f?(x) is a limit.
• Thus f?(x) is defined only when this limit
  exists.

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Example

• At right is the graph of a function f .
• We want to use this graph to sketch the graph of
  the derivative f?(x) .

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Solution

• We can estimate f?(x) at any x by
• drawing the tangent at the point (x, f(x)) and
• estimating its slope.
• Thus, for x 5 we draw the tangent at P in
  Fig. 2(a) (on the next slide), and estimate
  f?(5) 1.5 .
• Then we plot P?(5, 1.5) on the graph of f? .
• Repeating gives the graph in Fig. 2(b).

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Solution (contd)
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Solution (contd)
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Remarks on the Solution

• The tangents at A , B , and C are horizontal,
  so
• the derivative is 0 there, and
• the graph of f? crosses the x-axis at A?,
  B?, and C, directly beneath A, B, and C.
• Between
• A and B , f?(x) is positive
• B and C , f?(x) is negative.

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Example

• For the function f(x) x3 x ,
• Find a formula for f?(x)
• Compare the graphs of f and f?
• Solution On the
• next slide, we show that f?(x) 3x2 1
• following slide, we give the graphs of f and
  f? side-by-side

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Solution (contd)
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Solution (contd)

• Notice that f?(x) is
• zero when f has horizontal tangents, and
• positive when the tangents have positive slope

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Example

• Find f?(x) if
• Solution We use the definition as follows

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Solution (contd)
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Other Notations

• Here are common alternative notations for the
  derivative
• The symbols D and d/dx are called
  differentiation operators because they indicate
  the operation of differentiation, the process of
  calculating a derivative.

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Other Notations (contd)

• The Leibniz symbol dy/dx is not an actual
  ratio, but rather a synonym for f?(x) .
• We can write the definition of derivative as
• Also we can indicate the value f?(a) of a
  derivative dy/dx as

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Differentiability

• We begin with this definition
• This definition captures the fact that some
  functions have derivatives only at some values of
  x , not all.

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Example

• Where is the function f(x) x
  differentiable?
• Solution If x gt 0 , then
• x x and we can choose h small enough that
  x h gt 0 , so that x h x h
• Therefore

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Solution (contd)

• This means that f is differentiable for any x
  gt 0 .
• A similar argument shows that f is
  differentiable for any x lt 0 , as well.
• However for x 0 we have to consider

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Solution (contd)

• We compute the left and right limits separately
• Since these differ, f?(0) does not exist.
• Thus f is differentiable at all x ? 0 .

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Solution (contd)

• We can give a formula for f?(x)
• Also, on the next slide we graph f and f?
  side-by-side

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Solution (contd)
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Differentiability and Continuity

• We can show that if f is differentiable at a ,
  then f is continuous at a .
• However, as our preceding example shows, the
  converse is false
• The function f(x) x
• is continuous everywhere, but
• is not differentiable at x 0 .

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Failure of Differentiability

• A function can fail to be differentiable at x
  a in three different ways
• The graph of f can have a corner at x a
• as does the graph of f(x) x
• f can be discontinuous at x a
• The graph of f can have a vertical tangent line
  at x a .
• This means that f is continuous at a but
  f?(x) has an infinite limit as x ? a .
• We illustrate each of these possibilities

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Corner at x a
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Discontinuity at x a
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Vertical Tangent at x a
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More on Differentiability

• The next slides illustrate another way of looking
  at differentiability.
• We zoom in toward the point (a, f(a))
• If f is differentiable at x a , then the
  graph
• straightens out and
• appears more and more like a line.
• If f is not differentiable at x a , then no
  amount of zooming makes the graph linear.

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f Is Differentiable At a
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f Is Not Differentiable At a
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The Second Derivative

• If f is a differentiable function, then
• its derivative f? is also a function, so
• f? may have a derivative of its own, denoted
  by (f?) f?? , and called the second
  derivative of f .
• In Leibniz notation the second derivative of y
  f(x) is written

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Example

• If f(x) x3 x , find and interpret f??(x) .
• Solution We found earlier that the first
  derivative
• f?(x) 3x2 1 .
• On the next slide we use the limit definition of
  the derivative to show that
• f??(x) 6x

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Solution (contd)
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Solution (contd)

• On the next slide are the graphs of f , f? ,
  and f?? .
• We can interpret f??(x) as the slope of the
  curve y f?(x) at the point (x , f?(x)) .
• That is, f??(x) is the rate of change of the
  slope of the original curve y f(x) .
• Notice in Fig. 11 that
• f??(x) lt 0 when y f?(x) has a negative
  slope
• f??(x) gt 0 when y f?(x) has a positive
  slope.

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Solution (contd)
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Acceleration

• If s s(t) is the position function of a
  object moving in a straight line, then
• its first derivative gives the velocity v(t) of
  the object
• The acceleration a(t) of the object is the
  derivative of the velocity function, that is, the
  second derivative of the position function

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Example

• A car starts from rest and the graph of its
  position function in shown on the next slide.
• Here s is measured in feet and t in seconds.
• Use this to graph the velocity and acceleration
  of the car.
• What is the acceleration at t 2 seconds?

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Position Function of a Car
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Solution

• By measuring the slope of the graph ofs f(t)
  at t 0, 1, 2, 3, 4, and 5, we plot the
  velocity function v f?(t) (next slide).
• The acceleration when t 2 is a f??(2)
• the slope of the tangent line to the graph of
  f? when t 2 .
• The slope of this tangent line is

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Velocity Function
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Acceleration Function

• In a similar way we can graph a(t)

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Third Derivative

• The third derivative f???? is the derivative of
  the second derivative f???? (f??)? .
• If y f(x) , then alternative notations for the
  third derivative are

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Higher-Order Derivatives

• The process can be continued
• The fourth derivative f????? is usually denoted
  by f(4) .
• In general, the nth derivative of f is
• denoted by f(n) and
• obtained from f by differentiating n times.
• If y f(x) , then we write

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Example

• If f(x) x3 6x , find f????(x) and f(4)(x)
  .
• Solution Earlier we found that f???(x) 6x .
• The graph of y 6x is a line with slope 6
• Since the derivative f????(x) is the slope of
  f???(x) , we have
• f????(x) 6 for all values of x .
• Therefore, for all values of x ,
• f(4)(x) 0

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Review

• The derivative as a function
• The graph of f?? derived from the graph of f
• Finding formulas for f?(x)
• Differentiability
• Definition
• Differentiability implies continuity
• but not conversely
• Higher-order derivatives
• Homework
• 3-11odd, 12, 13, 19, 21, 25, 31, 33, 38, 39