Derivatives of exponential and logarithmic functions calculated.
We do not yet have a shortcut formula for the derivative of the natural logarithm, so let’s start from the definition. Set .
That is, The limit inside the logarithm is a bit beyond what we can deal with right now, so unless we can come up with a different strategy, we’re stuck.
What do we know about this logarithm? We know that the natural logarithm function is the inverse of the exponential function That is, Since we’re trying to find the derivative of , that means we’re trying to find . Rather than working with the logarithmic version of , let’s try to work with its exponential version . We’ll start by taking the derivative of both sides The right-hand side is easy, but what about the left-hand side? If we think of as just a function of , then the left-hand side is the exponential with the replaced by a function. It’s a Chain Rule problem, when we think of as having an outside function and an inside function . By Chain Rule, . Let’s put all this together.
We notice once again that , so . This gives our derivative formula.
We have found derivative formulas for the natural exponential function and the natural logarithm function , but we have not yet explored other bases. That will be our focus for the rest of the section.
For exponentials, we remember that any number can be written in the form for some specific value of . To determine the , we solve the equation so . That is, .
To find we are finding , which we know by Chain Rule.
Rewriting as we find our derivative formula.
To deal with logarithms of other bases, we rely on the change of base formula:
This formula allows us to replace a logarithm with one base with a logarithm with whatever base we want. There is one base that we like more than the rest, base . This means .
This gives our derivative formula.