# derivatives

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• Sep 9th 2006, 07:04 AM
shirel
derivatives
my homework..
• Sep 9th 2006, 08:02 AM
Jones
There are rules/formulae for these these kinds of operations.

The addition formula, the product rule etc...
Just use them.
• Sep 9th 2006, 01:26 PM
CaptainBlack
Quote:

Originally Posted by Jones
There are rules/formulae for these these kinds of operations.

The addition formula, the product rule etc...
Just use them.

He is being asked to prove them

RonL
• Sep 9th 2006, 01:34 PM
CaptainBlack
Quote:

Originally Posted by shirel
my homework..

1. consider the sum of two functions s(x): =f(x) + g(x) as a new function. calculate the derivative of s(x) and prove thereby that it equals the sum of the derivatives of g and g.

Consider:

$\displaystyle \frac{s(x+h)-s(x)}{h}=\frac{f(x+h)+g(x+h)-f(x)-g(x)}{h}$

$\displaystyle =\frac{f(x+h)-f(x)}{h}+\frac{g(x+h)-g(x)}{h}$

So:

$\displaystyle \lim_{h \to 0}\frac{s(x+h)-s(x)}{h}=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h} + \lim_{h \to 0}\frac{g(x+h)-g(x)}{h}$

which if all of these limits exist proves that:

$\displaystyle s'(x)=f'(x)+g'(x)$.

The others are similar, just use the definition of what a derivative is.
Replace the function by its other representation and rearrange to the
form we know it must have.

RonL

The meaning of this will become clear (as will my TeX errors) when the
LaTeX system is reinstalled:)
• Sep 9th 2006, 02:28 PM
Soroban
Hello, shirel!

#3 asks us to derive the Product Rule.
. . Too bad that Latex is off-line right now.

Quote:

3. Consider the product of two functions $\displaystyle p(x) = f(x)\cdot g(x)$
Calculate the derivative of $\displaystyle p(x)$.

We will derive: .$\displaystyle p'(x)\:=\:\lim_{h\to0}\frac{p(x+h) - p(x)}{h}$

We have: .$\displaystyle [(x+h) - p(x)\;=\;f(x+h)\cdot g(x+h) - f(x)\cdot g(x)$

Subtract and add $\displaystyle f(x+h)\cdot g(x):$

$\displaystyle p(x+h) - p(x)\;=\;f(x+h)\cdot g(x+h) - f(x+h)\cdot g(x) - f(x)\cdot g(x) + f(x+h)\cdot g(x)$

$\displaystyle p(x+h) - p(x)\;=\;f(x+h)\left[g(x+h) - g(x)\right] + g(x)\left[f(x+h) - f(x)\right]$

Divide by $\displaystyle h:$

[tex]\frac{p(x+h) - p(x)}{h} \;= \;f(x+h)\left[\frac{g(x+h)-g(x)}{h}\right] + g(x)\left[\frac{f(x+h)-f(x)}{h}\right] [/ math]

Take the limit:

$\displaystyle p'(x)\;=\;\lim_{h\to0}f(x+h)\cdot\lim_{h\to0}\left[\frac{g)x+h) - g(x)}{h}\right] + g(x)\cdot\lim_{h\to0}\left[\frac{f(x+h)-f(x)}{h}\right]$

Therefore: .$\displaystyle p'(x)\;=\;f(x)\cdot g'(x) + g(x)\cdot f'(x)$

• Sep 9th 2006, 04:19 PM
ThePerfectHacker
There is just one problem Soroban.
Unlike, CaptainBlank who states that the limits exists and thus does their sum, you did not. In front of mathematicians your proof would not be acceptable. (This is the major difference between Calculus and Analysis).
• Sep 10th 2006, 04:46 AM
Soroban
Hello, TPHacker!

Silly me . . . I thought this was a Calculus I problem.

. . where Existence must be established first.

I won't make that mistake again . . .

• Sep 10th 2006, 06:25 AM
ThePerfectHacker
Quote:

Originally Posted by Soroban
Hello, TPHacker!

Silly me . . . I thought this was a Calculus I problem.

I do not know if you are joking or not but there is no way this can be a Calculus I problem. The professor is not going to make you prove the product rule, which Leibniz was struggling to prove. I assume the students were familiar with the infromal proof from Calculus I but wanted them to show existence of limits. (Honestly how many people can spot that add-subtract trick).
• Sep 10th 2006, 09:25 AM
Plato
Oh, come Perfect! It was proved in very Calculus I class I ever taught. It is a standard way of illustrating that the ‘derivative limit’ does not have to be written in the order we all are used to seeing.

BTW: It usually goes, “There are 10 kinds of people: those who understand binary numbers and those who don’t.
• Sep 10th 2006, 09:30 AM
ThePerfectHacker
Quote:

Originally Posted by Plato
Oh, come Perfect!

Come where?

Quote:

Originally Posted by Plato
It was proved in very Calculus I class I ever taught.

Maybe, you taught an advanced class. Because I realized teachers do not like to get too theoretic about limits, they treat them more intuitevly.

I understand what you are saying that the proof is not so difficult, yes, I agree. I am saying that you need to show the two functions are differenciable and use rules about the existence of limits otherwise the proof is not sound.

Quote:

Originally Posted by Plato
BTW: It usually goes, “There are 10 kinds of people: those who understand binary numbers and those who don’t.

I know, that is the joke.
• Sep 10th 2006, 10:19 AM
topsquark
Quote:

Originally Posted by ThePerfectHacker
Maybe, you taught an advanced class. Because I realized teachers do not like to get too theoretic about limits, they treat them more intuitevly.

I understand what you are saying that the proof is not so difficult, yes, I agree. I am saying that you need to show the two functions are differenciable and use rules about the existence of limits otherwise the proof is not sound.

As it happens, when I did my undergrad proofs like these were usually saved for the "Advanced Calculus" class in the fourth semester.

-Dan
• Sep 10th 2006, 10:27 AM
Plato
Quote:

Originally Posted by topsquark
As it happens, when I did my undergrad proofs like these were usually saved for the "Advanced Calculus" class in the fourth semester.

But in an advanced calculus class I would hope that you gave epsilon-delta proof for these properties. Not simple limit proofs. Now there I do agree: epsilon-delta proofs have no place in beginning calculus.
• Sep 10th 2006, 10:32 AM
topsquark
Quote:

Originally Posted by Plato
But in an advanced calculus class I would hope that you gave epsilon-delta proof for these properties. Not simple limit proofs. Now there I do agree: epsilon-delta proofs have no place in beginning calculus.

Actually, I never took the class so I can't say for sure. But I believe you are right about it.

-Dan
• Sep 10th 2006, 10:45 AM
ThePerfectHacker
Quote:

Originally Posted by Plato
But in an advanced calculus class I would hope that you gave epsilon-delta proof for these properties. Not simple limit proofs. Now there I do agree: epsilon-delta proofs have no place in beginning calculus.

Once you have extablished the basic rules of limits through delta-epsilon you can stop using it for more advanced limit proofs.
• Sep 10th 2006, 04:12 PM
Plato
Quote:

Originally Posted by ThePerfectHacker
Once you have extablished the basic rules of limits through delta-epsilon you can stop using it for more advanced limit proofs.

Not sure what you point is? But I can tell you that over a thirty year stint of teaching real analysis, undergraduate and graduate, we never used anything but the epsilon-delta proofs. Limits, continuity, derivatives, and integrals are all defined using epsilon-deltas. We insist, well at least the school of mathematics I represent, a solid proof using epsilon and deltas is the only correct proof.
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