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Proof of Various Derivative Facts

Proof of Various Derivative Facts - Proof of Various...

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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,

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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.

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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.

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