5. Derivative of the Logarithmic Function
by M. Bourne
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Derivative of y = ln x
Derivative of a log of a function
Derivative of logs with base other than e
First, let's look at a graph of the log function with base e. The tangent at x = 2 is included on the 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).
We can conclude that the derivative of the logarithm function y = ln x is 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:
You will see it written in a few other ways as well. The following are equivalent:
If y = ln x, then
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
Example 2
Find the derivative of
y = ln x2
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:
where u' is the derivative of u
Another way to write this is
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:
Example 3
Find the derivative of
y = 2 ln (3x2 − 1).
Example 4
Find the derivative of
y = ln(1 − 2x)3.
Here it is in LiveMath:
Normal answer:
Example 5
Find the derivative
of 
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:
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:
[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.
Example 7
Find the derivative of y = 3 log7(x2 + 1).
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.
2. Find the derivative of
y = ln(cos x2).
3. Find the derivative of
y = x ln3 x.
4. Find the derivative of
3 ln xy + sin y = x2.
5. Find the derivative of
y = (sin x)x
by first taking logarithms of each side of the equation.
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