I just realised I don't really understand what some of these things mean. for example, we are just told to wtite dy/dx. I'm guessing this means something like "the change in y over the change in x, as the change in x tends to zero" or something.
but then we are told to write as an integral "dx" after the thing to be integrated, and that this means "with respect to x". which doesn't make a lot of sense.
Then a day or two ago I saw people doing things like:
dy/dx = 5x
dy = 5x dx
and things like that. is this allowed? can you multiply through by dx?
what do dx and dy mean?
With things like implicit differentiation and integration by parts still to be learnt I think it would help that I actually understood what I was doing with thr notation.
the integral dx probably means something like
$^(a) _(b) f(x) dx [thats the integral from a to b of f(x)]
sum f(x) according to the change in x (ie the limits)
thats why you cant integrate if you have
$^(u = a) _(u = b) f(x) dx
theres no change in the limits for any integration to be performed; no link between x and u... see?
techically i dont think you can multiply thru by dx but.. im not sure what the precise reason for how it all works is.
with implicit, you cant differentiatiate wrt the wrong variable so
d/dx(y) becomes d/dy(y) * dy/dx (by the chain rule)
we can differentiate the first part; its just 1
so this just becomes dy/dx (we just cant go any further)
yes, but dy/dx doesn't mean "with respect to y over with respect to x"
i was in the same situation last year, i didn't know what the teacher was going on about when he said with respect to. I though respect means a totally different thing lol.
This year i worked it out by reading around the topic.
dy = 5x dx
it's simple algebra, nothing complicated in it. It just looks complex.
dy/dx is just the gradient of a point in the curve.
d^2y/dx^2 is just differentiating twice.
Oh, it just means differentiating y with respect to x. if it was other way around i think it would be 1 over it. ocr MEI p2 explains it thoroughly i think.
its funny how everyone first takes the dy/dx as gospel and then when you see like dx/dt you flip out because you don't know what this means. calculus is a process of self discovery haha
what does f'(x) mean?
does it mean dy/dx of f(x)