The Derivative of a Logarithm
The Derivative of a Logarithm
If f(x) = loga x, then
d
1
log a x 
dx
 ln a  x
Notice if a = e, then
d
1
ln x 
dx
x
The Derivative of a Logarithm
If f(x) = loga g(x) and g(x) is differentiable, then
g ' x
d
log a  g ( x)  
dx
 ln a   g  x 
Notice if a = e, then
g ' x
d
ln  g ( x)  
dx
g  x
Examples
Find the derivative of each function:
a  10 & g  x   x
a.
f  x   log10 x
d
dx
1
x

f ' x 
 ln10   x x  ln10 
b. h  x   ln
g 'u  
d
dx

sin x
 ln e  
sin x
sin x

1
2

 sin x 
a  e & g  x   sin x
1 2
sin x
cos x
cos x
1

 cot x
2sin x
2
Example 2
Find the derivative of:

y  ln  x  1  2 x  9 
7
y  ln  x  1  ln  2 x  9 
7
You CAN use
Logarithm Laws to
expand to simplify
finding the derivative.
y  7  ln  x  1  ln  2 x  9 
Now take the derivative
y'  7
1
x 1
y'  7
2 x 9
 x 1 2 x 9 
y' 


2
2 x 9
16 x  65
 x 1 2 x 9 

2 x 1
 x 1 2 x 9 
Example 3
Find the derivative of:
Rewrite as a
piece-wise
function.
y' 
y' 
y  ln 2 x  1
ln  2 x  1 , x  0.5
y
ln  2 x  1 , x  0.5
Now take the derivative of each piece
2
2 x 1
2
2 x 1
y' 
Equal
y' 
2
2 x 1
2
2 x 1
Example 3 (Generalized)
Find the derivative of:
Rewrite as a
piece-wise
function.
y  log a u
log a  u  , x  c
y
log a  u  , x  c
Now take the derivative of each piece
y' 
u '
 ln a   u 
y' 
u'
 ln a u
y' 
u'
 ln a u
Equal
y' 
u'
 ln a u
The derivative of
each piece will
always be equal
The Derivative of a Logarithm
Composed with an Absolute Value
If f(x) = loga │g(x)│ and g(x) is differentiable, then
g ' x
d
log a g ( x) 
dx
 ln a   g  x 
Notice if a = e, then
g ' x
d
ln g ( x) 
dx
g  x
Ignore the Absolute Value.
White Board Challenge
Is the function below differentiable at x = 0?
e , x  0
f  x   2
 x  4 x, x  0
4x
No,
lim f  x   1 but lim f  x   0
x 0
x 0
It is not continuous at x  0.
Example 4
Find the
 x 12  2 x 2 3
If it is very
derivative of:
complicated or
2
x 1
impossible…
2
2
 x 1 2 x 3 Take the natural log of both sides to
expand the complicated quotient/product.
x 2 1
2
2
1
2
Now take the derivative of both sides.
ln f  x   ln


f  x 
ln f  x   2 ln  x  1  ln  2 x  3  ln  x  1
f ' x 
f  x
 2 
f ' x  2  
f ' x 


2
x 1
1
x 1
1
x 1
 
 
4x
2 x 2 3
4x
2 x 2 3

2x
Solve for f '
2
x 1
2x
1
2 x 2 1
2
2
x

1
2
x
 3 




 2 x42x3  x2x1 

1
2
 f  x
x 2 1


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'.
Example 5
Find the derivative of:
x

ln y  ln  sin x  


ln y  x  ln  sin x 
y   sin x 
x
Take the natural log
of both sides.
x is in the base and
exponent, so power
and exponential
rules do not apply.
Now take the derivative of both sides.
 x
 ln  sin x 
y '   x cot x  ln  sin x   y
y'
y
cos x
sin x
Solve for y '
y '   x cot x  ln  sin x    sin x 
x
1982 AB Free Response 5