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 keymethod 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 keymethods 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 handfull 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 (keymethods) 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 problemsolving.
Any answers appreciated, Thanks.
Links to examples:
https://tutorial.math.lamar.edu/classes/calci/DerivativeProofs.aspx
https://socratic.org/calculus/basicdifferentiationrules/proofoftheproductrule
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 keymethod 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 keymethods 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 handfull 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 (keymethods) 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 problemsolving.
Any answers appreciated, Thanks.
Links to examples:
https://tutorial.math.lamar.edu/classes/calci/DerivativeProofs.aspx
https://socratic.org/calculus/basicdifferentiationrules/proofoftheproductrule
Attachments

839.9 KB Views: 12

534.2 KB Views: 12