Hi,

I was wondering what the reasoning is behind the Chain rule and the Product rule - why they work rather than how. Also, what is integration by parts? I would really appreciate some help on this, as I can't get ahold of any good books on the subject.

Thank you,
Olof.

Ok, the chain rule. You start by differentiating y in terms of x: yes, that is dy/dx. Now, consider u, and u is a function of x, say u(x). We can differentiate u in terms of x, thats du/dx and y in terms of u, thats dy/du.

OK, so what does that achieve? So, what is dy/dx? It's the rate of change of y to x in terms of x. So dy/du is the rate of change of y to u in terms of u, which, if you write u in terms of x becomes the rate of change of y to u in terms of x. And du/dx is the rate of change of u to x in terms of x. So isn't dy/du xdu/dx the rate of change of y to x in terms of x, or dy/dx.

Ok that's the chain rule. I think I have a nice explanation on the product rule and possibly integration by parts in my room. So, I'll send you those when I get a chance to get back to the computer rooms...

Thanks Susan!

Consider this diagram:

Thanks again Susan. Couple of questions:
Does lim(dx -> 0) mean the limit of as dx tends to 0?
Also, what happens to the + du(dv/dx) part?

Thanks in advance,
Olof.

As du tends to 0, because dv/dx is a finite value, call it x, 0 x x=0. So it tends to 0 and disappears. You are right about limits.

Brad

Ah I see, cheers Brad.

u v= ó õ x x0  d(u v) dx dx

= ó õ x x0  du dx v dx+ x0 x   u dv dx dx

so

ó õ x x0  u dv dx dx=u v- ó õ x x0  du dx v dx

Does that help?

Susan

Cheers guys!

Regards,
Olof