Derivatives of logarithmic functions are mainly based on the chain rule. However, we can generalize it for any differentiable function with a logarithmic function. The differentiation of log is only under the base but we can differentiate under other bases, too.
Now we will prove this from first principles:
From first principles, .
Now let then
Find the derivative of at .
Solution 1: Use the chain rule.
Let and . Then we are asked to find .
Using chain rule, we know that
Since and we have
Solution 2: Use properties of logarithms.
We know the property of logarithms . Using this property,
If we differentiate both sides, we see that
since differentiation of which is a constant is
We have seen that , and this is the answer to this question.
Generalization: For any positive real number , we can conclude . Note that the derivative is independent of . This can be proven by writing instead of in the above solutions.
If is a positive real number and , then
We will use base-changing formula to change the base of the logarithm to
Since is a constant,
For any other type of log derivative, we use the base-changing formula.
Since this is a composite function, we can differentiate it using chain rule.
Now we will start with To find its derivative, we will substitute Now the derivative changes to So,
Find the derivative of .
Using the theorem, the derivative of is . In this problem, so .