5. Derivative of the Logarithmic Function

by M. Bourne


First, let's look at a graph of the log function with base e, which we write as y = ln x. The tangent at x = 2 is included on the graph.


ln graph

The slope of that tangent of y = ln x at x = 2 is 1/2. (We can observe this from the graph, by looking at the ratio rise/run).

If we were to find the slopes for some other values of x, we would find the following.

If y = ln x,

x 1 2 3 4 5
slope of graph 1 1/2 1/3 1/4 1/5
1/x 1 1/2 1/3 1/4 1/5

We see that the slope of the graph for each value of x is equal to 1/ x. This works for any positive value of x (we cannot have the logarithm of a negative number, of course).

If we did many more examples, we could conclude that the derivative of the logarithm function y = ln x is

dy/dx = 1/x.

Note 1: Actually, this result comes from first principles.

Note 2: We are using logarithms with base e. If you need a reminder about log functions, check out Log base e from before.


Derivative of the Logarithm Function y = ln x

The derivative of the logarithmic function y = ln x is given by:

d/dx(ln x)

You will see it written in a few other ways as well. The following are equivalent:

d/dx(loge x)

If y = ln x, then dy/dx = 1/x

For some problems, we can use the logarithm laws to simplify our log expression before differentiating it.

Example 1

Find the derivative of

y = ln 2x


Answer


Example 2

Find the derivative of

y = ln x2


Answer


Derivative of y = ln u (where u is a function of x)

Unfortunately, we can only use the logarithm laws to help us in a limited number of logarithm differentiation question types.

Most often, we need to find the derivative of a logarithm of some function of x. For example, we may need to find the derivative of y = 2 ln (3x2 − 1).

We need the following formula to solve such problems.

If

y = ln u

and u is some function of x, then:

u prime / u
where u' is the derivative of u

Another way to write this is

d/dx (ln u) = 1/u du/dx

You might also see the following form. It means the same thing.

If

y = ln f(x),

then the derivative of y is given by:

f prime on f

Example 3

Find the derivative of

y = 2 ln (3x2 − 1).


Answer


Example 4

Find the derivative of

y = ln(1 2x)3.


Here it is in LiveMath:

LIVEMath

Normal answer:


Answer


Example 5

Find the derivative of math expression


Answer


Differentiating Logarithmic Functions with Bases other than e

If

u = f(x) is a function of x,

and

y = logb u is a logarithm with base b,

then we can obtain the derivative of the logarithm function with base b using:

d/dx (log x)

where

u' is the derivative of u

logbe is a constant. See change of base rule to see how to work out such constants on your calculator.)

Note 1: This formula is derived from first principles.

Note 2: If we choose e as the base, then the derivative of ln u, where u is a function of x, simply gives us our formula above:

u prime / u

[Recall that logee = 1.]

[See the chapter on Exponential and Logarithmic Functions base e if you need a refresher on all this.]

Example 6

Find the derivative of y = log26x.


Answer


Example 7

Find the derivative of y = 3 log7(x2 + 1).


Answer


Note: Where possible, always use the properties of logarithms to simplify the process of obtaining the derivatives.


Exercises

1. Find the derivative of 

y = ln(2x3 x)2.


Answer



2. Find the derivative of 

y = ln(cos x2).


Answer



3. Find the derivative of 

y = x ln3 x.


Answer



4. Find the derivative of

3 ln xy + sin y = x2.


Answer



5. Find the derivative of

y = (sin x)x

by first taking logarithms of each side of the equation.


Answer




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