Derivatives of logarithms summary
Derivative of
The derivative of
We can prove this by using the fact that
Which gives the derivative stated above.
Now let’s check out some problems where we use this derivative:
Example
Let
Solution
We begin by noticing that
Example
Let
Solution
There are actually two ways of doing this problem, the first of these uses the chain rule, and the second one uses log rules. Let’s take a look at them both.
For the first we begin by looking at the function as it was given to us and notice that this is a function composition with inner function
Alternatively we can use log rules to rewrite
In this case we can differentiate immediately using the linearity of the derivative (i.e. that the constant 3 just pulls through the derivative). This gives us:
Now you try it!
Practice Problems for derivatives logarithms
Like learning by solving problems? Try this and more in the Dogl Calculus app 😃
Derivative of
The derivative of
This can be computed from the derivative of
Logarithm Change of Base Formula
Proof
If we take
Then using the power rule for logarithms
We using the log rule above we can rewrite
Now the
Example
Let
Solution
We can start by noticing that
What are logarithms again?
The logarithm function
Example
What is
Solution
This is just another way of asking the question: What power do we need to raise
Example
What is
Solution
As in the previous question, this is another way of asking the question: What power do we need to raise
In this case we know that
In addition to defining
Logarithms and Exponentials
The logarithmic function
Check for yourself if the final two identities in the box above make sense with the definition in terms of “the exponent that gives you
Graphs of logarithms
The relationship between logarithms and exponentials gives us a good way to look at graphs of logarithms. In particular we can find the graph of
Now you might be wondering why derivatives of exponentials are exponentials, but derivatives of logarithms don’t give back logarithms so nicely in the same way. Looking at the graph, and we can see immediately at least one problem, which is that the graph of the logarithm changes sign, but it’s slope is always either positive (when
Now let’s finish off with a quick look at what happens when the base
which means that