Why Are Mathematical Proofs So Confusing?

[Matt Jacques]

I understand how to use such things as product rules, quotient rules, parts by integration, but it bothers me I don't really have a deeper understanding of it.

My book offers rather rigorous proofs, they are all pretty much: assume this to be this and let this be that so it must equal this. Hmm. Ya.

Does anyone know of any good sites that proves them in understandable and no omitted steps?

 

[matt grime]

I'm sorry, I don't understand. How can your book simultaneously be too rigorous, yet omit steps? It may perhaps omit obvious calculations - it is up to you to put them back in if you can't see them.If you want motivation as to why the product rule is true, is this either too rigorous, or omitting too many steps?

f is diffible at x if

f(x+h) = f(x) +hf'(x) +h*o_1(h)

where o(h) is something that tends ot zero as h tends to zero. f' is defined to be the derivativethen
f(x+h)g(x+h)=(f(x)+hf'(x)+h*o_1(h))(g(x)+hg'(x)+h*o_2(h))

multiply out:f(x)g(x)+ h(f(x)g'(x)+f'(x)g(x)+ h*o_3(x))

where o_3(x) = hg'(x)f'(x)+hf'(x)o_2(x)+hg'(x)o_1(x)+ho_1(x)o_2(x)+f(x)o_2(x)+g(x)o_1(x)

which is a function that tends to zero as h tends to zero.

Hence the derivative of f(x)g(x) is f'(x)g(x)+f(x)g'(x)similar analysis allows you to prove the chain rule (messy) and the quotient rule, which actually just follows from the previous two. You should prove the chain rule - a liberal disrespect for the quantities you manipulate is to be encouraged.

 

[M98Ranger]

I completely understand your frustration with not fully understanding the proofs for various mathematical concepts. While it may seem like a daunting task, having a deeper understanding of these proofs can greatly enhance your overall understanding and application of these rules in problem-solving.

One suggestion I have is to seek out additional resources beyond your textbook. There are many online resources, such as Khan Academy and Mathisfun, that offer step-by-step explanations and visual aids for understanding proofs. You can also try searching for specific topics on YouTube, as there are many helpful videos created by math enthusiasts and educators.

Additionally, you may want to consider reaching out to your professor or a tutor for additional support. They can offer personalized explanations and help clarify any confusion you may have about the proofs.

Lastly, don't be afraid to break down the proofs and try to understand each step on its own. It may also be helpful to work through examples and practice problems to solidify your understanding.

Remember, understanding the proofs is just as important as knowing how to use the rules. Keep seeking out resources and don't give up, and you will eventually have a deeper understanding of these fundamental concepts in mathematics.