The Tangent Line Problem and The Area Problem (p. 101)

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2.8 Implicit Differentiation
Definition. We will say that a given equation in
x and y defines the function f implicitly if the
graph of y f(x) coincides with a portion of the
graph of the equation.

• Example
• The equation implicitly
  defines functions

• The equation implicitly defines the
  functions

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Two differentiable methods
There are two methods to differentiate the
functions defined implicitly by the equation.
For example
One way is to rewrite this equation as
, from which it follows that
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Two differentiable methods
The other method is to differentiate both sides
of the equation before solving for y in terms of
x, treating y as a differentiable function of x.
The method is called implicit differentiation.
With this approach we obtain
Since ,
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Implicit Differentiation
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Example
Example Use implicit differentiation to find dy
/ dx if
Solution
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Example
Example Find dy / dx if
Solution
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2.10 Logarithmic Functions
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Logarithm Function with Base a
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Natural Logarithm Function
Logarithms with base e and base 10 are so
important in applications that Calculators have
special keys for them.
logex is written as lnx
log10x is written as
logx The function ylnx is called the natural
logarithm function, and ylogx is Often called
the common logarithm function.
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Properties of Logarithms
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Properties of ax and logax
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Derivative of the Natural Logarithm Function
Note
Example
Solution
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Example
Example
Solution
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Derivatives of au
Note that
Example
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Derivatives of logau
Note that
Example
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The Number e as a Limit
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2.11 Inverse Trigonometric Functions
The six basic trigonometric functions are not
one-to-one (their values Repeat periodically).
However, we can restrict their domains to
intervals on which they are one-to-one.
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Six Inverse Trigonometric Functions

• Since the restricted functions are now
  one-to-one, they have inverse, which we denoted
  by
• These equations are read y equals the arcsine of
  x or y equals arcsin x and so on.
• Caution The -1 in the expressions for the
  inverse means inverse. It does
• Not mean reciprocal. The reciprocal of sinx is
  (sinx)-11/sinxcscx.

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Derivative of y sin-1x
Example Find dy/dx if Solution
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Derivative of y tan-1x
Example Find dy/dx if Solution
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Derivative of y sec-1x
Example Find dy/dx if Solution
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Derivative of the other Three
There is a much easier way to find the other
three inverse trigonometric Functions-arccosine,
arccotantent, and arccosecant, due to the
following Identities It follows easily
that the derivatives of the inverse cofunctions
are the negatives of the derivatives of the
corresponding inverse functions.
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