properties of logarithms log b (mn)= log b m + log b n ex: log 4 (15)= log 4 5 + log 4 3 log b...
DESCRIPTION
Goals: Find derivatives of functions involving the natural logarithmic function. Use logarithmic differentiation to take derivatives of more complicated functions. Sect. 5-1 Derivatives of Natural Logarithmic FunctionsTRANSCRIPT
Properties of Logarithms logb(MN)= logbM + logbN Ex: log4(15)= log45 + log43 logb(M/N)= logbM – logbN Ex: log3(50/2)= log350 – log32
logbMr = r logbM Ex: log7 103 = 3 log7 10
5-1 The Natural Logarithmic Function 5-4 Exponential Functions 5-5 Bases other than e and Applications 5-3 Inverse Functions and their Derivatives 5-8 Inverse Trig Functions 7-7 Indeterminate Forms and L’Hopital’s Rule
Chapter 5 Derivatives of Transcendental Functions
Goals:• Find derivatives of functions involving the natural logarithmic function.• Use logarithmic differentiation to take derivatives of more complicated functions.
Sect. 5-1 Derivatives of Natural Logarithmic Functions
1lnd duudx u dx
Derivatives of Natural Logarithmic Function
ln ( )y f x
( )u f x
1lnd xdx x
0x
0u
1ln | |d xdx x
0x
Example 1: Find the derivative of f(x)= xlnx.
Solution: This derivative will require the product rule.
1f (x) x lnx 1x
lnx1(x)f
Example 2: Find the derivative of g(x)= lnx/x
Solution: This derivative will require the quotient rule.
2
1x lnx 1xg (x)
x
2xlnx1(x)g
Example 3: Find the derivative of . )ln()( 1xxf 2
Solution: Using the chain rule for logarithmic functions.
( ) xf xx
2
21
Derivative of the inside, x²+1
The inside, x²+1
YOU TRY…
1. Find the derivative of y = x²lnx .
3. Find [ln | cos |]d xdx
32. Find [ln( 4)]d xdx
y’ = x² + (lnx)(2x)
= x + 2xlnx = x(1+2lnx)
x1
Example 1: Solution
Example 4 Differentiate 632 21ln xxy
Solution: There are two ways to do this problem. One is easy and the other is more difficult.
The difficult way:
62 3 5 62 3 2 3
6 62 3 2 3
d x 1 x 2 x 1 6 x 2 3x x 2 2xdxyx 1 x 2 x 1 x 2
55 6 3 2 32 2 3 3
6 62 3 2 3
2x x 2 9x x 1 x 218x x 1 x 2 2x x 2y
x 1 x 2 x 1 x 2
3 3 3 4 2
2 72 3 2 32 3
2x 9x 9x x 2 2x 10x 9x 2 20x 18x 4xy
x 1 x 2 x 1 x 2x 1 x 2
632 2x1xy ln
The easy way: First simplify the log using some of the expansion properties.
2x6ln1xln2xln1xln2x1xlny 32632632
2
2 3
6 3x2xyx 1 x 2
3 2 2
2 3 3 2
Now get a common denominator.
2x x 2 6 3x x 1y
x 1 x 2 x 2 x 1
4
3 2
2x +18xx 2 x 1
x x
4 24 18
4 2
2 3
20x 18x 4xx 1 x 2
Example 5: Find the derivative of .
34ln
xxy
Expand to y ln x 4 ln x 3
3x1
4x1y'
Now get a common denominator and simplify.
x-3 x 4
x 4 x 3 x 4 x 3
7x 4 x 3
Ex 6: Compute the derivative of .)3ln()( xxf
Note:
Therefore, we cannot use the properties of logs to bring the exponent down as a coefficient!
Instead, we must use the Chain Rule.
21
)3ln( x
21
)3ln( x 21
)3ln( x
x
xxf33)3ln(
21)(' 2
1
)3ln(21
xx
YOU TRY…Find the derivative of each of the
following:
1.
2. f(x) = ln (x2 – 3)5
3. f(x) = [ln (x2 – 3)]5
4.
5.
1( ) ln2
xf xx
2 3
5 2
( 2)( ) ln1
xf xx x
3 1( ) ln5 2
xf xx
ln (x2 – 3)5 = 5 ln (x2 – 3)
= ln (3x +1) – ln (5x-2)
3 1ln5 2
xx
Solution 1
ddx
lnx 1
x 2
1x 1x 2
ddx
x 1
x 2
x 2x 1
x 2 1 (x 1) 12 (x 2) 1 2
x 2
x 2 1
2 (x 1)(x 1)(x 2)
x 5
2(x 1)(x 2)
Solution 4 Way 1
If we first simplify the given function using the laws of logarithms, then the differentiation becomes easier:
12
1ln ln( 1) ln( 2)2
1 1 11 2 2
d x d x xdx dxx
x x
Solution 4 Way 2
Solution 5
2 26 1 2
2 5( 1)y x xy xx x
dxdy
2 3
5 2
( 2)ln ln1
xyx x
2 21ln 3ln( 2) (ln ln 1 )5
y x x x
ClosureExplain how to differentiate the following function both the hard way and the easy way.
2 3
5 2
( 2)( ) ln1
xf xx x
Taking this derivative would involve the Power, Product, Quotient, and Chain (twice) Rules.
Instead we can use a technique called logarithmic differentiation to simplify the process.
3/4 2
5Differentiate: (
13 2)
x xyx
LOGARITHMIC DIFFERENTIATION
Take the “ln” of both sides and use properties of logarithms to expand.
Differentiate implicitly with respect to x.
2
1 3 1 1 2 354 2 3 21
dy xy dx x xx
23 14 2ln ln ln( 1) 5ln(3 2)y x x x
3/4 2
5Differentiate: (
13 2)
x xyx
Solve for dy / dx.
Since we have an explicit expression for y, we can substitute and write:
3 / 4 2
5 2
1 3 154 3 2(3 2) 1
dy x x xdx x xx x
2
3 154 1 3 2
dy xydx x x x
1. Take natural logarithms of both sides of an equation y = f(x) and use the Laws of Logarithms to simplify.
2. Differentiate implicitly with respect to x.
3. Solve the resulting equation for y’.4. Replace y with f(x).
STEPS IN LOGARITHMIC DIFFERENTIATION
Example 1: Differentiate . ))(( 1x1xxy 2
Solution: This problem could be easily done by multiplying the expression out.
))((lnln 1x1xxy 2
ln ln ln( ) ln( )y x x x 21 1y xy x x x
21 1 2
1 1
1x2x
1x1
x1
yy
2
1x
x21x
1x1yy 2
))(( 1x1xxy 2
22
1 1 2xy x x 1 (x 1)x x 1 x 1
12x3x4xy
2x2xxx1xxxy
1xx2x1)x(x1)(x1xy
1x1)(x1xx2x
1x1)(x1xx
x1)(x1xx
23
23323
22
2
222
y
Continue to simplify…
Let’s double check to make sure that derivative is correct by multiplying out the original and then taking the derivative.
1x2x3x4y
xxxx1xxxy1x1xxy
23
23422
2
)(
))((
Remember this problem was to practice the technique. You would not use it on something this simple.
Chain Rule and Product Rule
Ex 2. 2Find '( ) given ( ) ( 3) 5 4.h x h x x x 1
2 2( 3)(5 4)x x 1 1
2 2 21'( ) ( 3) (5 4) 5 (5 4) 22
h x x x x x
25( 3) 2 (5 4)
2 (5 4)x x x
x
This time use logarithmic differentiation to solve.
Chain Rule and Quotient Rule 72 1Find '( ) given ( ) .
3 5xG x G xx
6
2
3 5 2 2 1 32 1( ) 73 5 3 5
x xxG xx x
66
2 8
91 2 12 1 13( ) 73 5 3 5 3 5
xxG xx x x
Ex 3.
This time use logarithmic differentiation to solve.
3 24
11. Find 6 2 5
ddr r r
You Try…
3 2 42. Find given ( 3 ) (2 1) .dy y x x xdx
3. Find dy/dx if . 2 1/2( 1)( 3)
1x xy
x
1 2 1xdy x xdx
1ln ln 2 xy x
ln 1 ln 2y x x
1 11 ln 22
dy x xy dx x
1 ln 22
dy xy xdx x
1 12 ln 22
xdy xx xdx x
Logarithmic differentiation is also used when the variable is in the base and the exponent.
.)2( ateDifferenti :4Ex 1 xxy
)()( xgxfy
Example 5: Differentiate .
ln ln lnxy x x x
xy x
' 1 1(ln )2
y x xy x x
1 ln 2 ln'2 2
xx xy y xx x x
Ex 6: Find an equation of the tangent line
to the graph of at .,12
2sin
xy x
1. y = xx
You Try…
Differentiate each of the following functions.
2. y = (sinx)lnx
xxdxdy x ln1
ln y = ln xx
ln y = x ln x
xx
xdxdy
yln11
xydxdy ln1
xxdxdy x ln1
y = xx
ClosureExplain the steps for the different methods used to differentiate the following functions.1. y = x 4
2. y = xx