Proof of Various Derivative Facts/Formulas/Properties

Proof of Various Derivative Facts/Formulas/Properties

In this section we�re going to prove many of the various derivative facts, formulas and/or properties that we encountered in the early part of the Derivatives chapter. Not all of them will be proved here and some will only be proved for special cases, but at least you�ll see that some of them aren�t just pulled out of the air.

Theorem, from Definition of Derivative

Proof

Because is differentiable at we know that

exists. We�ll need this in a bit.

If we next assume that we can write the following,

Then basic properties of limits tells us that we have,

The first limit on the right is just as we noted above and the second limit is clearly zero and so,

Okay, we�ve managed to prove that . But just how does this help us to prove that is continuous at ?

Let�s start with the following.

Note that we�ve just added in zero on the right side. A little rewriting and the use of limit properties gives,

Now, we just proved above that and because is a constant we also know that and so this becomes,

Or, in other words, but this is exactly what it means for is continuous at and so we�re done.

Power Rule :

There are actually three proofs that we can give here and we�re going to go through all three here so you can see all of them. However, having said that, for the first two we will need to restrict n to be a positive integer. At the time that the Power Rule was introduced only enough information has been given to allow the proof for only integers. So, the first two proofs are really to be read at that point.

The third proof will work for any real number n. However, it does assume that you�ve read most of the Derivatives chapter and so should only be read after you�ve gone through the whole chapter.

Product Rule :

As with the Power Rule above, the Product Rule can be proved either by using the definition of the derivative or it can be proved using Logarithmic Differentiation. We�ll show both proofs here.

Quotient Rule :

Again, we can do this using the definition of the derivative or with Logarithmic Definition.

Chain Rule