The Derivative of a Logarithm. If f(x) = log a x, then Notice if a = e, then.
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Transcript of The Derivative of a Logarithm. If f(x) = log a x, then Notice if a = e, then.
![Page 1: The Derivative of a Logarithm. If f(x) = log a x, then Notice if a = e, then.](https://reader031.fdocuments.us/reader031/viewer/2022032312/56649de85503460f94ae226f/html5/thumbnails/1.jpg)
The Derivative of a Logarithm
![Page 2: The Derivative of a Logarithm. If f(x) = log a x, then Notice if a = e, then.](https://reader031.fdocuments.us/reader031/viewer/2022032312/56649de85503460f94ae226f/html5/thumbnails/2.jpg)
The Derivative of a Logarithm
If f(x) = loga x, then
1
loglna
dx
dx a x
Notice if a = e, then
1ln
dx
dx x
![Page 3: The Derivative of a Logarithm. If f(x) = log a x, then Notice if a = e, then.](https://reader031.fdocuments.us/reader031/viewer/2022032312/56649de85503460f94ae226f/html5/thumbnails/3.jpg)
The Derivative of a Logarithm
If f(x) = loga g(x) and g(x) is differentiable, then
'log ( )
lna
g xdg x
dx a g x
Notice if a = e, then
'
ln ( )g xd
g xdx g x
![Page 4: The Derivative of a Logarithm. If f(x) = log a x, then Notice if a = e, then.](https://reader031.fdocuments.us/reader031/viewer/2022032312/56649de85503460f94ae226f/html5/thumbnails/4.jpg)
Examples
Find the derivative of each function:
10. loga f x x
. ln sinb h x x
'
ln10
ddx x
f xx
1
ln10x
sin'
ln sin
ddx x
g ue x
1 212 sin cos
sin
x x
x
10 &a g x x
& sina e g x x
cos
2sin
x
x
1cot
2x
![Page 5: The Derivative of a Logarithm. If f(x) = log a x, then Notice if a = e, then.](https://reader031.fdocuments.us/reader031/viewer/2022032312/56649de85503460f94ae226f/html5/thumbnails/5.jpg)
Example 2
Find the derivative of: 7ln 1 2 9y x x
You CAN use Logarithm Laws to expand to simplify
finding the derivative.
7ln 1 ln 2 9y x x
7 ln 1 ln 2 9y x x Now take the derivative
1 21 2 9' 7 x xy
2 12 9
1 2 9 1 2 9' 7 xx
x x x xy
16 651 2 9
' xx x
y
![Page 6: The Derivative of a Logarithm. If f(x) = log a x, then Notice if a = e, then.](https://reader031.fdocuments.us/reader031/viewer/2022032312/56649de85503460f94ae226f/html5/thumbnails/6.jpg)
Example 3
Find the derivative of: ln 2 1y x Rewrite as a piece-wise function.
ln 2 1 , 0.5
ln 2 1 , 0.5
x xy
x x
Now take the derivative of each piece
22 1' xy
2
2 1' xy 2
2 1' xy
22 1' xy
Equal
![Page 7: The Derivative of a Logarithm. If f(x) = log a x, then Notice if a = e, then.](https://reader031.fdocuments.us/reader031/viewer/2022032312/56649de85503460f94ae226f/html5/thumbnails/7.jpg)
Example 3 (Generalized)
Find the derivative of: logay u
log ,
log ,
a
a
u x cy
u x c
Now take the derivative of each piece
'
ln' u
a uy
'
ln' u
a uy
'
ln' u
a uy
'
ln' u
a uy
Equal
The derivative of each piece will
always be equal
Rewrite as a piece-wise function.
![Page 8: The Derivative of a Logarithm. If f(x) = log a x, then Notice if a = e, then.](https://reader031.fdocuments.us/reader031/viewer/2022032312/56649de85503460f94ae226f/html5/thumbnails/8.jpg)
The Derivative of a Logarithm Composed with an Absolute Value
If f(x) = loga │g(x)│ and g(x) is differentiable, then
'log ( )
lna
g xdg x
dx a g x
Notice if a = e, then
'
ln ( )g xd
g xdx g x
Ignore the Absolute Value.
![Page 9: The Derivative of a Logarithm. If f(x) = log a x, then Notice if a = e, then.](https://reader031.fdocuments.us/reader031/viewer/2022032312/56649de85503460f94ae226f/html5/thumbnails/9.jpg)
White Board Challenge
Is the function below differentiable at x = 0?
0 0
, lim 1 lim 0
It is not continuous at 0.x x
No f x but f x
x
4
2
, 0
4 , 0
xe xf x
x x x
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2 2
2 2 2
1 2 342
1 2 3 1 1'
x xx x
x x x xf x
Example 4
Find the derivative of: 2 2
2
1 2 3
1
x x
xf x
Take the natural log of both sides to expand the complicated quotient/product. 2 2
2
1 2 3
1ln ln
x x
xf x
2 212ln 2 ln 1 ln 2 3 ln 1f x x x x
Now take the derivative of both sides. 2 2
' 4 21 11 22 3 1
2f x x xxf x x x
2 2
4 21 11 22 3 1
' 2 x xx x x
f x f x
Solve for f '
If it is very complicated or impossible…
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Logarithmic Differentiation
1. Take the natural logarithm of both sides.
2. Simplify the “x” side using the properties of logarithms.
3. Differentiate both sides of the equation.
4. Solve for y'.
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' cot ln sin sinx
y x x x x
Example 5
Find the derivative of: sinx
y xTake the natural log
of both sides. ln ln sinx
y x ln ln siny x x
Now take the derivative of both sides.
' cossin ln siny x
y xx x ' cot ln siny x x x y
Solve for y '
x is in the base and exponent, so power
and exponential rules do not apply.
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1982 AB Free Response 5