In this note, I'm going to demonstrate another way to prove the derivative formulas and using logarithmic differentiation. The most commonly taught method is with the definition of a derivative, but this can get ugly and can easily result in a stupid mistake.
First, let's take a look at the derivative of the natural logarithm:
If then Derive both sides with respect to So that is the derivative of If is a differentiable function in then by the chain rule, Now we can move to logarithmic differentiation.
Logarithmic differentiation is commonly used for really ugly functions to differentiate. You use it if you are told to derive for example. If you try to use the chain rule, product rule, and quotient rule for this, you are in for one heck of a ride, and the answer you come up with has a really good chance of being wrong. You can take the natural logarithm of both sides and apply the properties of logarithms and the chain rule to find the answer.
This is much easier to evaluate for values of If you are actually trying to expand an expression like this, then you are being trolled. A time where you might expand the expression is if you are deriving
So now onto the proofs. and are differentiable functions of
Let's start with the product rule. Let So this is the proof of the product rule. Now let's move on the quotient rule. Now let And now the quotient rule has been proven.
In my opinion, this is a lot easier than trying to figure out what you need to add and subtract to the expression found using the definition of a derivative. Hopefully you do too! Thanks for reading this post, and remember to occasionally check #TrevorsTips for tricks to solve problems.