Transcript Logarithmic Differentiation

Logarithmic Differentiation
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© Annie Patton
Aim of Lesson
To use Logarithms to
differentiate functions,
when the function has an
index which itself is a
function.
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© Annie Patton
Logarithmic Differentiation
• This method is used, when the function
has an index, which itself is a function.
• For example y=x2x or y=xsin x.
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© Annie Patton
y=x2x
Take the Natural Log of both sides
ln y= ln x 2x
ln y = 2 x ln x
d ( ln y) d (2 x ln x)

dx
dx
d (ln y) dy
1
 ln x.2+2x.
dy
dx
x
1 dy
 2 ln x  2
y dx
dy
=y(2lnx+2)
dx
dy
=x 2x (2lnx+2)
dx
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© Annie Patton
y=xsinx
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ln y=ln x
sin x
d ln y d sin x ln x

dx
dx
1 dy sin x

 ln x cos x
y dx
x
ln y=sin x ln x
d lny dy
1
= sin x +ln x cos x
dy dx
x
dy
sin x
 y(
 ln x cos x)
dx
x
dy
sin x
 x sin x (
 ln x cos x )
dx
x
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© Annie Patton
y=2x.x2
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x
ln y= ln 2 .x
2
ln y=ln 2  ln x
x
2
d ln y d ( x ln2 + 2 ln x)

dx
dx
ln y = x ln 2 + 2 ln x
d ln y d ln y dy
1

 x.0  ln 2.+2
dx
dy dx
x
1 dy
2
 ln 2 
y dx
x
dy
2
2
x 2
 y (ln 2  )  2 x (ln 2  )
dx
x
x
© Annie Patton
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ln x  ln e
y
x
y ln x  x ln e
d ln x
dy dx
y
 ln x  because ln e=1
dx
dx dx
1
dy
y
+ln x
=1
x
dx
dy
y
ln x
=1dx
x
y
1dy
x-y
x
=
=
dx
ln x
x ln x
© Annie Patton
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dy
If y=3 .x . Find
.
dx
x
3
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ln y  ln 2 .x  ln 2  ln x  x ln 2  3ln x
x
3
x
3
d (ln y ) d (ln y ) dy 1 dy


dx
dy dx y dx
1 dy
d (ln 3)
dx
d (ln x)
3
3
x
 ln 3  3
 0  ln 3   ln 3 
y dx
dx
dx
dx
x
x
dy
3
3
x
3
 y (ln 3  )  3 .x (ln 3  )
dx
x
x
© Annie Patton
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If y= x
lnx
. Find
dy
.
dx
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ln y  ln x
ln x
 ln x.ln x  (ln x)
2
d (ln y ) d (ln x) 2

dx
dx
1 dy d (ln x) 2 d (ln x)
1

 2 ln x
y dx d (ln x) dx
x
dy
2ln x
=x ln x
dx
x
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© Annie Patton
Homework. Differentiate the
following:
1. y=x
cosx
2. y=(cos x)
3. y=8
x
4. y=3
cosx
x
( x  2)
5. y 
2
( x  2)
3
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© Annie Patton
Logarithmic Differentiation
• This method is used,
when the function has
an index, which itself
is a function.
• For example y=x2x or
y=xsin x.
Take the Natural Log of both sides
For example y=22 x
ln y= ln x 2x
© Annie Patton