The Derivative of a Logarithm

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

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

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

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

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

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.

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.

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

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…

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'.

' 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.

1982 AB Free Response 5