Pauls Online Notes : Calculus I - Proof of Various Derivative Properties

Paul's Online Math Notes

Calculus I

You can navigate through this E-Book using the menu to the left. For E-Books that have a Chapter/Section organization each option in the menu to the left indicates a chapter and will open a menu showing the sections in that chapter. Alternatively, you can navigate to the next/previous section or chapter by clicking the links in the boxes at the very top and bottom of the material.

Also, depending upon the E-Book, it will be possible to download the complete E-Book, the chapter containing the current section and/or the current section. You can do this be clicking on the E-Book, Chapter, and/or the Section link provided below.

For those pages with mathematics on them you can, in most cases, enlarge the mathematics portion by clicking on the equation. Click the enlarged version to hide it.


Proof of Various Limit Properties	E-Book Chapter Section	Proof of Trig Limits

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

If is differentiable at then is continuous at .

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.

Proof of Sum/Difference of Two Functions :

This is easy enough to prove using the definition of the derivative. We’ll start with the sum of two functions. First plug the sum into the definition of the derivative and rewrite the numerator a little.

Now, break up the fraction into two pieces and recall that the limit of a sum is the sum of the limits. Using this fact we see that we end up with the definition of the derivative for each of the two functions.

The proof of the difference of two functions in nearly identical so we’ll give it here without any explanation.

Proof of Constant Times a Function :

This is property is very easy to prove using the definition provided you recall that we can factor a constant out of a limit. Here’s the work for this property.

Proof of the Derivative of a Constant :

This is very easy to prove using the definition of the derivative so define and the use the definition of the derivative.

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.

Proof 1

In this case as noted above we need to assume that n is a positive integer. We’ll use the definition of the derivative and the Binomial Theorem in this theorem. The Binomial Theorem tells us that,

where,

are called the binomial coefficients and is the factorial.

So, let’s go through the details of this proof. First, plug into the definition of the derivative and use the Binomial Theorem to expand out the first term.

Now, notice that we can cancel an and then each term in the numerator will have an h in them that can be factored out and then canceled against the h in the numerator. At this point we can evaluate the limit.

Proof 2

For this proof we’ll again need to restrict n to be a positive integer. In this case if we define we know from the alternate limit form of the definition of the derivative that the derivative is given by,

Now we have the following formula,

You can verify this if you’d like by simply multiplying the two factors together. Also, notice that there are a total of n terms in the second factor (this will be important in a bit).

If we plug this into the formula for the derivative we see that we can cancel the and then compute the limit.

After combining the exponents in each term we can see that we get the same term. So, then recalling that there are n terms in second factor we can see that we get what we claimed it would be.

To completely finish this off we simply replace the a with an x to get,

Proof 3

In this proof we no longer need to restrict n to be a positive integer. It can now be any real number. However, this proof also assumes that you’ve read all the way through the Derivative chapter. In particular it needs both Implicit Differentiation and Logarithmic Differentiation. If you’ve not read, and understand, these sections then this proof will not make any sense to you.

So, to get set up for logarithmic differentiation let’s first define then take the log of both sides, simplify the right side using logarithm properties and then differentiate using implicit differentiation.

Finally, all we need to do is solve for and then substitute in for y.