derivatives of logarithmic and exponential functions of logarithmic and exponential functions we...
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Derivatives of Logarithmic and Exponential Functions
We begin by finding the derivative of a logarithmic function directly from the definition and the properties of logarithms. We remind you of the following facts:
1. log log loga a cb b bc
= −
2. log ( ) logra r ab b
=
3. If f is a continuous function, and then lim ( ( )) ( ( ))f g x f g a
x a=
→
lim ( ) ( )g x g ax a
=→
( )11 14. lim 1 lim 1 where
0
xe v vvx xx v
= + = + = →∞ →
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Proof. log ( ) loglog lim
0
x h xd b bxbdx hh
+ −= →
( )1lim log 10
vbvxv
= +→
1lim log0
x hb xhh
+ = →
1lim log 10
hb xhh
= + →
(Definition)
(Fact 1)
(algebra)
(Substitute v = h/x)
Theorem. The derivative of f(x) = logb(x) is
And more generally
1(log )eb x
log ( ) 1log ( )d u dub e
bdx u dx
=
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( )11 lim log 1
0v vbx v
= + →
1 log ( )ebx
=
( )1 1lim log 10
vbx vv
= + →
( )11 log lim 1
0v vbx v
= + →
(Constant 1/x comes outside limit)
(Fact 2)
(Fact 3)
(Fact 4)
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Example. Find the derivative of
(a) (b) (c)
As a special case, when b = e, we have
Theorem. The derivative of f(x) = ln(x) is
It follows that
1 1(log )ee x x
=
This simple formula explains in part the use of e as a base for logarithms in mathematical work.
[ ]ln( ) 1d u dudx u dx
=
log( )1 log( )
xx+
ln(ln( ))x ln(2 )x+
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[ ]
[ ]
1 11 log( ) log( ) log( ) log( )log( )
21 log( ) 1 log( )
x e x ex x xx x
′ + − = + +
1 [ln( )] 1ln(ln( ))ln( ) ln( )
d xxx dx x x
′= =
(a)
(b)
[2 ]1ln(2 )2
d xxdxx
′ + + = + (c)
[ ] [ ]
1log( ) log( )
2 21 log( ) 1 log( )
e exx x x
= =+ +
1 1 12 2 4 2x x x x
= = + +
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Slope of tangent line is 1 at x = 1
Derivative of ln(x) is 1x
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Slope of tangent line at x = 2 is 1/2.
Derivative of ln(x) is 1x
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Slope of tangent line at x = 4 is 1/4.
Derivative of ln(x) is 1x
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If you are taking the derivative of the logarithm of a complicated expression involving primarily products, quotients, and powers , it is usually best to simplify the logarithm first before taking derivatives.
Example. Compute the derivative of the function3( 1) cos( )
( ) ln2
x xf x
x x
+=
−
We can write this derivative as but the computation
of the derivative of u will be very long and involved.
1 duu dx
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1( ) 3ln( 1) ln(cos( )) ln( ) ln(2 )2
f x x x x x= + + − − −So
Then( )
1 sin( ) 1 1 1( ) 31 cos( ) 2 2
xf xx x x x
− −′ = + − −+ −
3 1 1tan( )1 4 2
xx x x
= − − ++ −
Better Solution: First we simplify the function.
3( 1) cos( ) 3ln ln ( 1) ln(cos( )) ln( ) ln( 2 )2
x xx x x x
x x
+ = + + − − − −
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Logarithmic Differentiation
Even if a function does not involve a logarithm, it’s differentiation can often be simplified by using a technique called logarithmic differentiation. This technique should be used on functions that are primarily composed of products, quotients, and roots.
Example. Find the derivative of the function 3sin( )cos( ) tan ( )x x xy
x=
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1. We begin by taking the natural logarithm of both sides of the equation 3sin( )cos( ) tan ( )x x xy
x=
3sin( )cos( ) tan ( )ln( ) ln
x x xy
x
=
1ln(sin( )) ln(cos( )) 3ln(tan( )) ln( )2
x x x x= + + −
2. Now take the derivative of both sides, treating y as an implicit function of x on the left, getting
21 cos( ) sin( ) sec ( ) 113sin( ) cos( ) tan( ) 2
dy x x xy dx x x x x
−= + + −
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Thus
2sec ( ) 1cot( ) tan( ) 3
tan( ) 2x
y x xx x
= − + −
3 2sin( )cos( ) tan ( ) sec ( ) 1cot( ) tan( ) 3
tan( ) 2x x x x
x xx xx
= − + −
2cos( ) sin( ) sec ( ) 1 13
sin( ) cos( ) tan( ) 2x x xdy yx x x xdx
−= + + −
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Derivatives of arbitrary powers of x
Theorem. For any real number r, rational or irrational, we have
ln( ) ln( ) ln( )ry x r x= =Proof. We use logarithmic differentiation. Let y = xr. Then
1d r rx rxdx
−=
, therefore
Examples. 1d x x
dxπ ππ −=
5 5 15x x
′ −=
1 dy ry dx x
= 1rdy ry rx rrx
dx x x−= = =and so
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Derivatives of Exponential FunctionsWe can also use logarithmic differentiation to find the derivatives of the functions f(x) = bx.
Theorem.
Proof. Let y = bx. Then ln(y) = ln(bx) = xln(b). Therefore1 ln( )dy by dx
= and ln( ) ln( )xdy y b b b
dx= = . The second formula
Theorem.
follows by the chain rule. If b = e,we have:
and, more generally, d dux ux ue e e edx dx
′ = =
ln( ) and, more generally, ln( )x d dux u ub b b b b b
dx dx
′ = =
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Height is 1 at x = 1.
Slope of tangent line is 1 at x = 1
Derivative of exp(x) is exp(x)
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Slope of tangent lineHeight here
Derivative of exp(x) is exp(x)
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Height
Slope of tangent line
Derivative of exp(x) is exp(x)
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Example. Find the derivative of with respect to x.cos( )xe
Example. Find the derivative of with respect to x. 34
2x
Solution:
Solution:
[ ] [ ]cos( ) cos( ) cos( )cos( ) sin( )x x xe e x e x′ ′= =−
( )3 3 34 4 3 2 42 ln(2)2 4 ln(2) 12 2d x x xx xdx
′ = =
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We have now shown the following rules for differentiation.
Basic Rule Generalized Rule
11. log logd x eb bdx x=
1log logd duu eb bdx u dx=
12. ln( )d xdx x
= 1ln( )d duu
dx u dx=
3. ln( )d x xb b bdx
=
ln( )d duu ub b bdx dx
=
4. d x xe edx
=
d duu ue edx dx
=
15.
rd rx rxdx
− =
1rd duru rudx dx
− =
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We have now shown the following rules for differentiation.
Basic Rule Generalized Rule
11. log logd x eb bdx x=
1log logd duu eb bdx u dx=
12. ln( )d xdx x
= 1ln( )d duu
dx u dx=
3. ln( )d x xb b bdx
=
ln( )d duu ub b bdx dx
=
4. d x xe edx
=
d duu ue edx dx
=
15.
rd rx rxdx
− =
1rd duru rudx dx
− =
WARNING
Do not confuse 3 and 4 with 5
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Example. Find the derivative of with respect to x.3ln( )y x=
Solution1: 21 3 33
3 3dy d xxdx dx xx x
= = =
3ln( )y x=Solution2: so 1 33dydx x x
= =
Example. Find the derivative of with respect to x.3
ln( )y x=
Solution: 22 3ln( )3ln( ) ln( )dy d xx x
dx dx x= =
Miscellaneous Examples
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Example. Find the derivative of with respect to x.2sin (ln( ))y x=
Solution: 2sin(ln( ))cos(ln( ))2sin(ln( ))cos(ln( )) ln( )dy d x xx x x
dx dx x= =
Example. Find the derivative of with respect to x.2ln(sin )y x=
Solution: 2ln(sin( ), so2 cos( )sin( ) 2 2cot( )
sin( ) sin( )
y xdy d xx xdx x dx x
=
= = =
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Example. Find the derivative of with respect to x.
2 2ln 3( 3) ( 1)
x xy
x x
−=
+ −
Solution: First simplify to get
( )2 3ln( ) ln( 2 ) ln ( 3) ln( 1)y x x x x= + − − + − −12ln( ) ln(2 ) 3ln( 3) ln( 1)2
x x x x= + − − + − −
Then
12 1 3 122 3 1
yxx x x
− ′= + − − − + −
2 1 3 14 2 3 1x x x x
= − − −− + −
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Example. Find the derivative of with respect to x.3( )xx ey e −=
Solution: 3 33 3( ) ( ) 1 3x xdy d x xx e x ee x e e e
dx dx − −= − = −
Example. Find the derivative of with respect to x.( ) 22 1y x= +
Solution: ( )
2 1 2 12 2 22 1 1 2 1 2dy dx x x xdx dx
− − = + + = + 2 122 2 1x x
− = +
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Example. Find the derivative of with respect to x by using logarithmic differentiation.
Solution:
151
xx
−+
1Let 5 .1
xyx
−=+
1 11 1 15Then ln( ) ln ln ln( 1) ln( 1)1 15 5 5
x xy x xx x
− − = = = − − + + +
1 11 1 1 2so 1 1y 5 5 5( 1)( 1)
dyx xdx x x
= − = − + − +
2 1 255( 1)( 1) 1 5( 1)( 1)
xdy yx x x x xdx
− = = − + + − +
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Example. Find the derivative of with respect to xby using logarithmic differentiation.
Solution:
tan( )ln( ) xx
tan( )Let ln( ) xy x=
tan( )Then ln( ) ln ln( ) tan( )ln(ln( ))xy x x x = = 1 1 12so sec ( )ln(ln( )) tan( )y ln( )
dy x x xdx x x
= +
1 12sec ( ) ln(ln( )) tan( )ln( )
dy y x x xx xdx
= +
tan( )tan( ) 2ln( ) sec ( ) ln(ln( ))ln( )
xxx x xx x
= +