The Derivative

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The Derivative
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Definition
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Example (1)Find the derivative of f(x) 4 at
any point x
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Example (2)Find the derivative of f(x) 4x at
any point x
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Example (3)Find the derivative of f(x) x2 at
any point x
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Example (4)Find the derivative of f(x) x3 at
any point x
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Example (5)Find the derivative of f(x) x4 at
any point x
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Example (6)Find the derivative of f(x) 3x3
5x2 - 2x 7 at any point
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Questions

• Find from the definition the derivative of each
  of the following functions
• 1. f(x) vx
• 2. f(x) 1/x
• 3. f(x) 1/x2
• 4. f(x) 5 / (2x 3)

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Power Rule

• Let
• f(x) xn , where n is a real number other than
  zero
• Then
• f'(x) n xn-1
• If f(x) constant , then f'(x) 0

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Algebra of Derivatives
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Example (1)
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Solution
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The Chain RuleThe derivative of composite
functionfor the case f(x) gn(x)

• Let
• f(x) gn(x)
• Then
• f' (x) ngn-1(x) . g'(x)
• Example
• Let f(x) (3x8 - 5x 3 )20
• Then f(x) 20 (3x8 - 5x 3 )19 (24x7 - 5)

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Examples (1)
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Example (2)
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Solution
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Example (3)
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Homework
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Answers of Questions(1)

• Find from the definition the derivative of
• each of the following functions
• 1. f(x) vx
• 2. f(x) 1/x
• 3. f(x) 1/x2
• 4. f(x) 5 / (2x 3)

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4.
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Differentiability Continuity
1. If a function is differentiable at a point,
then it is continuous at that point. Thus if a
function is not differentiable at a point, then
it cannot be continuous at that point. But the
converse is not true. A function can be
continuous at a point without being
differentiable at that point.
2. A point of the graph at which the graph of the
function has a vertical tangent or a sharp corner
is a point where the function is not
differentiable regardless of continuity
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Examples(1) Sharp Corner
This function (Graph it!) is continuous at the
point x2, since the limit and value of the
function at that point are equal ( Show that!)
but it is not differentiable at that point, since
the right derivative of f at x2 is not equal to
the left derivative a that point.
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Examples(2) Vertical TangentWhen both right and
left derivatives are 8 or both are - 8
This function (Graph it!) is continuous at the
point x0, since the limit and value of the
function at that point are equal ( Show that!)
but it is not differentiable at that point, since
the right derivative (and also the left
derivatives) of f do not exist at x2 ( both are
8)
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Examples(3) CuspWhen the one of the one-sided
derivative is 8 and the other is- 8
This function (Graph it!) is continuous at the
point x0, since the limit and value of the
function at that point are equal ( Show that!)
but it is not differentiable at that point,
since at x2 the right derivative does not
exist ( is 8) and also the left derivatives
does not exist and is -8)
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Limits Involving Trigonometric Functions
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All trigonometric functions are continuous a each
point of their domains, which is R for the sine
cosine functions(?The limit of sinx and cosx at
any real number a are sina and cosa
respectively), R- p/2, - p/2, 3p/2, -
3p/2, for the Tangent and the Secant
functionsand R-0, p, - p, 3p, - 3p , for
the Cotangent and the Cosecant functions.
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Important Identity
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Examples (1)
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Example (2)
Solution
f is continuous for all x other than zero. To
check, whether it is continues, as well at x0,
we need to that its limit at x0 is equal to
f(0), which is given as zero.
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Example (3)
Solution
For f to be continuous at x0, we need its
limit at x 0 to exist and to equal the value at
that point, which is 9. Since its right limit at
x0 is equal also 9, it remains that its left
limit at that point be equal to that value.
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Example (4)
Solution
All members are continuous for all x other than
2. For a member to be continuous at x2, we need
the limit of the member function at x 2 to exist
and to equal the value at that point, which is
1/2c. Since the right limit at x2 of any member
is equal o 1/2c, it remains that its left limit
at that point be equal to that value.
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