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The Natural Logarithmic The Natural Logarithmic Functions: DifferentiationFunctions: DifferentiationDifferentiation, implicit differentiation, tangent lines.
So from the definition of the ln function, we should be able to figure out the derivative of the ln function. Part 2 just follows from the chain rule.
Ex. 3 p. 326 Differentiation of Logarithmic Functions
. ln 3dx
da x (u = 3x)
' 3
3
u
u x
1
x
3. ln 5dx
db x
3(u = 5 )x 2
3
' 3
5
u x
u x
. lndx
dc x x Product rule! ln ln
d dx x x xdx dx
1
ln 1x xx
1 ln x
4. ln
dx
dd x
Chain Rule!
3 14 ln
xx
3
4 ln
x
x
Ex 4 p. 236 Logarithmic Properties as an aid to differentiation.
Differentiate ( ) ln 3 5f x x
Rewrite before differentiating: 1( ) ln 3 5
2f x x
1 3'( )
2 3 5f x
x
3
2 3 5x
Ex 5. p327 Logarithmic Properties as aids to differentiating
Differentiate 23
4
1( ) ln
2 1
x xf x
x
Rewrite! 3 41( ) ln 2ln( 1) ln(2 1)2f x x x x
2 3
3 4
1 3 1 8'( ) 2
1 2 2 1
x xf x
x x x
2 3
3 4
1 6 4
1 2 1
x x
x x x
This would have been very difficult without the rewrite!
Sometimes it is useful or even necessary to use logarithmic aids in non-logarithmic problems.Ex 6 p. 327 Logarithmic Differentiation
Differentiate3
4
( 5), x 5
6
xy
x
Take natural log of BOTH sides3
4
( 5)ln ln
6
xy
x
Expand right side41ln 3ln( 5) ln( 6)
2y x x
Differentiate implicitly3
4
' 1 1 43
5 2 6
y x
y x x
Simplify and solve for y’
3
4
3 2'
5 6
xy y
x x
3 4 3
44
( 5) 10 18'
5 66
x x xy
x xx
Simplify
3 4 4 3
44
( 5) 3 18 2 10
5 66
x x x x
x xx
2 4 3
34 2
( 5) 10 18'
6
x x xy
x
Example where it is necessary to use logarithmic differentiation!
Differentiate
32ln ln( 1)
xy x Take ln of both
sides 3 2ln( 1)x x
32( 1)
xy x
Product rule, differentiate implicitly
3 2 22
' 2ln( 1) 3
1
y xx x x
y x
Simplify, solve for y’4
2 22
2' 3 ln( 1)
1
xy y x x
x
3 42 2 2
2
2' ( 1) 3 ln( 1)
1
x xy x x x
x
3 312 4 2 2 2' ( 1) (2 ) ( 1) 3 ln( 1)x x
y x x x x x
Substitute in y
Whew!
In more abstract form,
d/dx (u^v) = (u^v)(d/dx (v ln u)) = u^v(vu'/u + v'ln u))
as discovered by one of our students!
I think he should submit for publication and get his name on a theorem.
Because the natural log is undefined for negative numbers, we often encounter functions defined asy = ln |u|.
You can disregard the absolute value symbols and differentiate as usual!
See proof on p. 328 if interested.
Ex 7 p. 328 Derivative involving absolute value
Derive ( ) ln sinf x x
' cos'( )
sin
u xf x
u x cot x
Ex. 8 p. 328 Finding relative extrema
Locate relative extrema of 3ln 3 5 , 2y x x x
First, differentiate y, 2
3
3 3'
3 5
xy
x x
2
3
3( 1)
3 5
x
x x
then find critical numbers.
Critical numbers of numerator are x = 1, -1.From the denominator, x ≈ -2.2790188, which is not in domain.
Applying first derivative test, in the interval [-2, -1), y’ > 0 so increasing on that interval. In the interval (-1, 1),y’ < 0 so decreasing on that interval and point (-1, ln 7) is Rel. Max. In interval (1, ∞), y’ > 0 so point (1, ln 3) is a Rel. Min.
You can do anything with natural log functions that you can do with other kinds of functions, so you can find equations of tangent lines at some point, equations of normal lines at some point, relative extrema, inflection points, etc.
3ln 3 5y x x
5.1b p. 329/ 43-87 EOO, do 77,81,93,95,104-105
