We shall first look at the value of :
Now we find the derivative of using the formal definition of the derivative:
Let . Note that as approaches , approaches 0. So we can redefine our limit as:
Here we could take the natural logarithm outside the limit because it doesn't have anything to do with the limit (we could have chosen not to). We then substituted the value of .
If we wanted, we could go through that same process again for a generalized base, but it is easier just to use properties of logs and realize that:
Since 1 / ln(b) is a constant, we can just take it outside of the derivative:
Which leaves us with the generalized form of:
We shall take two different approaches to finding the derivative of . The first approach:
The second approach:
Note that in the second approach we made some use of the chain rule.Thus:
so that we have proved the following rule:
Now that we have derived a specific case, let us extend things to the general case. Assuming that a is a positive real constant, we wish to calculate:
One of the oldest tricks in mathematics is to break a problem down into a form that we already know we can handle. Since we have already determined the derivative of ex, we will attempt to rewrite ax in that form.
Using that eln(c) = c and that ln(ab) = b · ln(a), we find that:
Thus, we simply apply the chain rule:
We can use the properties of the logarithm, particularly the natural log, to differentiate more difficult functions, such a products with many terms, quotients of composed functions, or functions with variable or function exponents. We do this by taking the natural logarithm of both sides, re-arranging terms using the logarithm laws below, and then differentiating both sides implicitly, before multiplying through by y.
See the examples below.
We shall now prove the validity of the power rule using logarithmic differentiation.
- Example 2
Suppose we wished to differentiate
We take the natural logarithm of both sides
Differentiating implicitly, recalling the chain rule
Multiplying by y, the original function
- Example 3
Let us differentiate a function
Taking the natural logarithm of left and right
We then differentiate both sides, recalling the product and chain rules
Multiplying by the original function y
- Example 4
Take a function
We then differentiate
And finally multiply by y