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Transcript of DERIVATIVES OF FUNGTION
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MATHEMATICS 2FOR SENIOR HIGH SCHOOL
CHAPTER 8
DERIVATIVES OF FUNCTIONS
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Derivatives of Algebraic Function
If the function of is differentiable for each in the domain ofD where D R,
then the derivative of the function and states by formula of:
a. is called the derivative of the function of
b. The process of finding from is called differential.
c. Another notation for the derivative of the function is or or
The notation of or is called the Notation of leibniz
)(xf x
)('xf
)(xf
h
xfhxf
xfh
)()(
)(' lim0
)('xf )(xf
)(xfy dxxdf )(y dx
dy
dx
xdf )(
dx
dy
1.Derivative definition
A
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2.Derivatives of Exponential Function
Let = where n is a called an exponential function. The derivative of this kind
of function is :
)(xfnax
1
1
12321
332221
0
332221
0
0
0
)1,(
)...)3,()2,()1,((lim
.. .)3,()2,()1,(lim
).. .)3,()2,()1,((lim
)(lim
)()(
lim
n
n
nnnn
nnnn
h
nnnnnn
h
nn
h
h
anx
xnaC
h
hhhxnChxnCxnCa
h
hhxnChxnChxnCa
h
xhhxnChxnChxnCxa
h
axhxa
h
xfhxf
)(xf
Therefore, if = where n is a positive integer, then =)(xf nax )(xf 1nanx
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3. Derivatives of Sums and Differences of Function
Suppose where is the sum of two function , if and
exist, then:
Similar to the proof of
If , then
)(xf )()( xvxu )(xf )(xv )(xu
)(xv
)()(
)()()()(
)()()()(
))()(())()((
)()(
limlim
lim
lim
lim
00
0
0
0
xvxu
h
xvhxu
h
xuhxu
hxvhxv
hxuhxu
h
xvxuhxvhxu
h
xfhxf
hh
h
h
h
)(xf
)(xf )()( xvxu )(xf )()( xvxu )(xf )()( xvxu
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4.Derivatives of Products and Quotients of Function
Suppose where and exist, then:)(xf )()( xvxu )(xu )(xv
)()()()()()()()(
)()()()()(
)()()(
)()()(
)()()()()()()()(
))()(())()((
)()(
limlimlimlim
lim
lim
lim
lim
0000
0
0
0
0
xvxuxvxuxuxvxvxu
h
xuhxu
h
xvhxvhxu
h
xuhxuxv
h
xvhxvhxu
h
xvxuxvhxuxvhxuhxvhxu
h
xvxuhxvhxu
h
xfhxf
hhhh
h
h
h
h
)(xf
Therefore, if , then)(xf )()( xvxu )(xf )()()()( xvxuxvxu
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1. Derivatives of Sine and Cosine Functions
Suppose = sin ,then:
Similarly, the derivatives of the function = cos is -sin
)(xf x
x
x
h
h
hx
h
hhx
h
xhx
h
xfhxf
hh
h
h
h
cos
2
1cos2
2
1sin
)2(2
1cos2
2
1sin)2(
2
1cos2
)sin()sin(
)()(
limlim
lim
lim
lim
00
0
0
0
)(xf
)(xf x )(xf x
Derivatives of Trigonometric Functions
B
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2.Derivatives of Tangent Functions
Using the derivates of sine and cosine function, we can determine the derivatives of
other tigonometric functions. Next is the explanation of deriving tangen functions.
Suppose : then
then
0cos,cos
sintan)( x
x
xxxf
,cos)(
,sin)(
xxv
xxu
xxv
xxu
sin)(
cos)(
xxx
xx
x
xxxx
xv
xvxuxvxu
2
22
22
2
2
seccos
1
cos
sincos
)(cos
)sin(sincoscos
)(
)()()()(
)(xf
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From the result, we can draw conclusions about the derivatives of trigonometric
functions is as follows
If = sin , then = cos
If = cos , then = -sin
If = tan , then =
If = cot , then =
If = sec , then = sec tan
If = csc , then = -scs cot
)(xf)(xf)(xf
)(xf
)(xf
)(xf
x
xx
x
x
x
)(xf)(xf
)(xf
)(xf)(xf
)(xf
x2sec
xx
xx x
x
x2csc
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Derivatives of Composition Function
Definition
If f and g both are differentiable and F = f o g is a composition function defined by
F(x) = f(g(x)), then F can be differentiated to be F with the formula of
In the notation of Leibniz, ify = f(u) and u = g(x) and both function are differentiable
the derivative of the function value can be formulated is as followers.)())(()()()( xgxgfxgfxF
dxdu
dudy
dxdy
C
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Equation of tangent on a curve
h
afhaf
ahaafhaf
)()(
)()(
ABm
The tangent to a curve
The gradient of the curve at point A represents
The derivative of the functionsy = f(x) atx = a.The gradient of the tangent on curve ofy = f(x) at
Point (a,f(a)) is determined by:
The gradient of the chord AB is:)( ABm
)()()(
lim0
afh
afhafm
h
And can be determined by the formula of:
)(11 xxmyy where )( 1xfm
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Increased and Decreased Functions
a. The function of is called increased function if > 0
b. The function of is called decreased function if < 0
)(xf)(xf
)(xf)(xf
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The Theorems of Increased and Decreased Functions
If a function ofy = f(a) is continuous and differentiable atx = a andf (a) = 0, then
the function has a stationary value atx = a
a. A condition of a function will have a stationary value iff (x) = 0 for
a value ofx.
b. If a function of has stationary valuef(a) at x = a, then point of(a,f(a)) is
called a stationary point.
)(xf
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Stationary ValuesTypes of stationary values
Maximum returning point Minimum returning point
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Horizontal turning value
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Application of Derivatives
If function off(x) is differentiated and result in another new function off (x)
then f (x) is called the second derivative of f(x) . The notation of the first derivative
is y = f(x) isy or f (x) or or , the notation of the second
derivative isy = f(x) the similar ways:dx
dy
dx
df
2
2
dx
yd2
2
dx
fdyorf (x) or or
