2.2 Fundamental derivation rules

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2.2 Fundamental derivation rules

• 1.Derivation rules for sum, difference, product
  and quotient of functions
• 2. Derivation rules for composite functions
• 3. The derivative of an inverse function
• 4.Derivatives for the fundamental elementary
  functions
• 5.Higher order derivatives

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1.Derivation rules of rational operations
Th2.2.1
3
proof
, then
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(2)
then
proof Suppose
( C is a constant )
corollary
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(1) (2) are also valid for any finite number of
functions.
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(3)
then
Proof Suppose
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Example 2.2.2-3
?
Similarly
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2. Derivation rules for composite functions
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The theorem can be generalized to several
intermediate variables.
For instance,
Key Decompose the function into some simple
functions and find derivatives from outside to
inside successively.
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Example
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• P122 T4(4)

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3. The derivative of an inverse function
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Example 2.2.8. Find the derivatives of the
inverse trigonometric functions.
Solution For
then
by
similarly
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4.Derivatives for the fundamental elementary
functions
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Ex . Suppose
where exists, find
Solution
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Definition 2.2.1.
5.Higher -order derivatives
is derivable,
If the derivative of
then its derivative is called the second order
derivative of
Denoted by
or
i.e.
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In general, the derivative of order n-1
derivative is called the derivative of order n,
and is denoted by i.e.
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Example 1.
find
Suppose
Solution
By means of mathematical induction ,
we have
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Example2. Suppose
find
Solution
In particular
find
Example3. Suppose
Solution
0 ! 1
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Example4. Suppose
find
Solution
so ,
Bu the similar means
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The operation rules of higher-order
derivatives(Th2.2.4)
are both derivable of order n , then
Leibniz formula
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P123T11 Let
find the highest order n such
Solution
while
but
does not exists.
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The method of finding higher order derivatives
(1) Find the derivative order by order
(2) By means of mathematical induction
(3) By Leibniz formula
(4) From the known derivative formula
For instance,
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Example P123T10(4)
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Exercise in class
1. How to find the derivative of order n for the
following functions?
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2.3 Derivation of implicit functions and
functions defined by parametric equations
1.Derivation of implicit functions
2.Method of derivation of a function defined by
parametric equations
3.Related rates of change
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1.Derivation of implicit functions
explicit function.
If the function is defined by the
equation
then it is called an implicit function .
For instance,
can define a function with the dependent
variable y and independent variable x ,but it
can not be explicited.
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• How to find the derivative of an implicit
  function?

Taking derivative on both sides with respect to x
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Example. Find the tangent line of
at
Solution Taking derivatives on both sides
respect to x
So the tangent line is
i.e.
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The method of logarithmic derivation.(P133T3)
Example Find the derivative of
Solu Taking the natural logarithm on both sides
Taking derivative on both sides respect to x
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Note
1) For a powerexponent function
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2) Some explicit functions .
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and,
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2.Method of derivation of a function defined by
parametric equations
If the parametric equation
can define a function
is derivable and
yf(x),
then
If we have
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If
(here means x is the function of y )
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if
are both twice derivable,
and
then
is twice derivable.
,we have
From the new parametric equation
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?
Note
If
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Example. Suppose that
can define the function
find
Solution Taking derivatives on both sides
so
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3.related rates of change
are two derivable functions
are dependent
are dependent too
is called the related rates of change.
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• Solving the problem of related rates of change

Find the equation F(x,y)0
Taking derivative on both sides respect to t
Get the expression between
Find the rate that we want
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Have a think Suppose
find the derivative of the inverse function .
Solution
1.
Solution2