Proof. Definition of the product rule

Thread Starter

ZimmerJ

Joined Dec 9, 2020
55
Hello, this is more a general question regarding proof. I can't think of a specific problem but such as definitions of rules in mathematics and how someone would go about proving a given definition . I'm fairly new and this is one of the things i have struggled with through out my courses.

A good example i came across is the product rule for derivatives, which you can prove in different ways. I will include two pictures which displays two ways on how to do it. In these pictures i marked what seems to be the key-method with red.

In my book there is another way by using the rate of change of area for a quadrant, but i don't fully understand it so i won't try to explain it.

Question:
In the pictures i included, the publishers refer to these key-methods as "algebraic trick" and "manipulation". How do you get good at this? Is there a technique that you can use or will it come by itself naturally after a lot of grinding solving such problems? Or are there just a hand-full of these methods that i can keep in a formula sheet? (Not necessarily rules per se, but for specific problems as well)

Many of the harder questions not only in the book but also during tests, requires this. Sure, i can probably learn some of these methods and they will apply to some definitions and problems, but certainly not all. I am not sure how to express my confusion, but i feel that this type of problem solving (key-methods) in algebraic terms is the major thing i lack when approaching problems. If i can learn a good technique that maybe some of you use, it would make me much more dynamic/fluent in problem-solving.

Any answers appreciated, Thanks.

Links to examples:
https://tutorial.math.lamar.edu/classes/calci/DerivativeProofs.aspx
https://socratic.org/calculus/basic-differentiation-rules/proof-of-the-product-rule
 

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Papabravo

Joined Feb 24, 2006
17,316
Hello, this is more a general question regarding proof. I can't think of a specific problem but such as definitions of rules in mathematics and how someone would go about proving a given definition . I'm fairly new and this is one of the things i have struggled with through out my courses.

A good example i came across is the product rule for derivatives, which you can prove in different ways. I will include two pictures which displays two ways on how to do it. In these pictures i marked what seems to be the key-method with red.

In my book there is another way by using the rate of change of area for a quadrant, but i don't fully understand it so i won't try to explain it.

Question:
In the pictures i included, the publishers refer to these key-methods as "algebraic trick" and "manipulation". How do you get good at this? Is there a technique that you can use or will it come by itself naturally after a lot of grinding solving such problems? Or are there just a hand-full of these methods that i can keep in a formula sheet? (Not necessarily rules per se, but for specific problems as well)

Many of the harder questions not only in the book but also during tests, requires this. Sure, i can probably learn some of these methods and they will apply to some definitions and problems, but certainly not all. I am not sure how to express my confusion, but i feel that this type of problem solving (key-methods) in algebraic terms is the major thing i lack when approaching problems. If i can learn a good technique that maybe some of you use, it would make me much more dynamic/fluent in problem-solving.

Any answers appreciated, Thanks.

Links to examples:
https://tutorial.math.lamar.edu/classes/calci/DerivativeProofs.aspx
https://socratic.org/calculus/basic-differentiation-rules/proof-of-the-product-rule
As near as I can tell you can't. You either have the creativity and insight to do this sort of thing or you don't. Many people get by on sheer memory of techniques they have seen. I've known maybe two people who could do this kind of thing on problems they had never seen before.

The same thing happens with integration. Some people have an insight or an intuition about how to do these things and the rest of us struggle, or buy a copy of Abramowitz & Stegun
 
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Thread Starter

ZimmerJ

Joined Dec 9, 2020
55
As near as I can tell you can't. You either have the creativity and insight to do this sort of thing or you don't. Many people get by on sheer memory of techniques they have seen. I've known maybe two people who could do this kind of thing on problems they had never seen before.

The same thing happens with integration. Some people have an insight or an intuition about how to doe these things and the rest of us struggle, or buy a copy of Abramowitz & Stegun
Hm, i was thinking that most people who work/study in math-heavy fields such as physics are good at this. Maybe this is not true and i shouldn't feel to bad about it?

I had no idea of such books, thanks!
 

Papabravo

Joined Feb 24, 2006
17,316
Hm, i was thinking that most people who work/study in math-heavy fields such as physics are good at this. Maybe this is not true and i shouldn't feel to bad about it?

I had no idea of such books, thanks!
In a career spanning half a century I spent very little time on developing proofs. I did apply them like a set of blocks to construct new things that had never been built before. I think the time I spent as a kid taking things apart and putting them back together was about as valuable a set of experiences as anything else I picked up along the way.
 

bogosort

Joined Sep 24, 2011
678
In the pictures i included, the publishers refer to these key-methods as "algebraic trick" and "manipulation". How do you get good at this? Is there a technique that you can use or will it come by itself naturally after a lot of grinding solving such problems? Or are there just a hand-full of these methods that i can keep in a formula sheet?
The "algebra tricks" used to rewrite and simplify equations have been developed over hundreds of years. You learn them by doing lots of algebraic manipulations (solving problems), and -- as with anything else -- you get better at it the more you do it.

Of course, there are general principles involved. Algebraic manipulation is mostly about using identities (equivalent expressions) that let us rewrite something complicated into an equivalent but simpler form. You've undoubtedly seen and used identities such as \[ \frac{a}{a} = 1 \] for any nonzero 'a', and \[ \frac{a+b}{c} = \frac{a}{c} + \frac{b}{c} \] for any nonzero 'c'. Using these and other identities, along with the rules of commutativity, distributivity, and associativity of terms, we can often rewrite expressions in simpler forms: \[ \begin{align} \frac{x-7}{x-1} &= \frac{x - 1 -6}{x-1} \\ &= \frac{(x-1) - 6}{x-1} \\ &= \frac{x-1}{x-1} - \frac{6}{x-1} \\ &= 1 - \frac{6}{x-1} \end{align} \] With practice you'll start to "see" complicated expressions in terms of identities that will simplify them. In my case, two classes in particular helped me develop manipulation intuition: trigonometry (which is essentially all about finding identities) and differential equations (the techniques of which are the algebra of calculus). Don't feel bad if a textbook uses a "trick" that you don't immediately understand. Analyze the trick and try to find the general principles it used. The next time you see a similar "trick", you'll recognize it, and soon you'll be able to use it yourself.

I'd also point out that algebraic manipulation, though it can be useful in certain types of proof, is itself not a mathematical proof. Proving theorems is an entirely different thing, and can be high art when done well. If you're in school and interested in learning to prove ideas, I highly recommend taking a discrete mathematics course. That will teach you propositional and predicate logic, set theory, and basic proof techniques.
 

Thread Starter

ZimmerJ

Joined Dec 9, 2020
55
The "algebra tricks" used to rewrite and simplify equations have been developed over hundreds of years. You learn them by doing lots of algebraic manipulations (solving problems), and -- as with anything else -- you get better at it the more you do it.

Of course, there are general principles involved. Algebraic manipulation is mostly about using identities (equivalent expressions) that let us rewrite something complicated into an equivalent but simpler form. You've undoubtedly seen and used identities such as \[ \frac{a}{a} = 1 \] for any nonzero 'a', and \[ \frac{a+b}{c} = \frac{a}{c} + \frac{b}{c} \] for any nonzero 'c'. Using these and other identities, along with the rules of commutativity, distributivity, and associativity of terms, we can often rewrite expressions in simpler forms: \[ \begin{align} \frac{x-7}{x-1} &= \frac{x - 1 -6}{x-1} \\ &= \frac{(x-1) - 6}{x-1} \\ &= \frac{x-1}{x-1} - \frac{6}{x-1} \\ &= 1 - \frac{6}{x-1} \end{align} \] With practice you'll start to "see" complicated expressions in terms of identities that will simplify them. In my case, two classes in particular helped me develop manipulation intuition: trigonometry (which is essentially all about finding identities) and differential equations (the techniques of which are the algebra of calculus). Don't feel bad if a textbook uses a "trick" that you don't immediately understand. Analyze the trick and try to find the general principles it used. The next time you see a similar "trick", you'll recognize it, and soon you'll be able to use it yourself.

I'd also point out that algebraic manipulation, though it can be useful in certain types of proof, is itself not a mathematical proof. Proving theorems is an entirely different thing, and can be high art when done well. If you're in school and interested in learning to prove ideas, I highly recommend taking a discrete mathematics course. That will teach you propositional and predicate logic, set theory, and basic proof techniques.
This is great, thank you for the feedback. Exactly, seeing those identities that will actually simply a rather complicated expression seems to be the hard part. Usually i wonder off and come to a dead end, not really knowing where i am trying to go. I will keep in mind what you said and practice.

Yes i have seen it a lot in trigonometry, though differential equations is right ahead for me so that will be interesting. I will definitely check that course out, sounds like it will benefit a lot.

Thanks again!
 

dcbingaman

Joined Jun 30, 2021
498
Hello, this is more a general question regarding proof. I can't think of a specific problem but such as definitions of rules in mathematics and how someone would go about proving a given definition . I'm fairly new and this is one of the things i have struggled with through out my courses.

A good example i came across is the product rule for derivatives, which you can prove in different ways. I will include two pictures which displays two ways on how to do it. In these pictures i marked what seems to be the key-method with red.

In my book there is another way by using the rate of change of area for a quadrant, but i don't fully understand it so i won't try to explain it.

Question:
In the pictures i included, the publishers refer to these key-methods as "algebraic trick" and "manipulation". How do you get good at this? Is there a technique that you can use or will it come by itself naturally after a lot of grinding solving such problems? Or are there just a hand-full of these methods that i can keep in a formula sheet? (Not necessarily rules per se, but for specific problems as well)

Many of the harder questions not only in the book but also during tests, requires this. Sure, i can probably learn some of these methods and they will apply to some definitions and problems, but certainly not all. I am not sure how to express my confusion, but i feel that this type of problem solving (key-methods) in algebraic terms is the major thing i lack when approaching problems. If i can learn a good technique that maybe some of you use, it would make me much more dynamic/fluent in problem-solving.

Any answers appreciated, Thanks.

Links to examples:
https://tutorial.math.lamar.edu/classes/calci/DerivativeProofs.aspx
https://socratic.org/calculus/basic-differentiation-rules/proof-of-the-product-rule
I like this one taken from geometry:

https://web.mit.edu/wwmath/calculus/differentiation/products.html

It is easier to visualize the product rule as a rectangle ab with da and db differentials, both differentials are multiplied by the sides of the rectangle for a*db+b*da with the small da*db rectangle vanishing away.
 
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