1) i don't think it is intuition.it is just integration by substitution.
if you say that some t=log(x)
then dt/dx=1/x
or dt=dx/x
d(\log x)=\frac{dx}{x}
now you can substitute dx/x with d(log(x))
(1) Why is it OK to change the integrand's step like this:
What is the (geometrical or some other) intuition behind this?
This is from a (somewhat) rigorous analysis book ( - proofs galore), and such integration-step-substitutions were being made without explanation.
(2) Let's say that . Please help me write with differentials:
How do you continue?
sure!
2)
You get that from the chain rule. Chain rule - Wikipedia, the free encyclopedia
edit: i fixed the whole thing right up.
Got slightly further: since , we can rewrite
To put everything in one place:
Is this going in the right direction? If so, then what is if you have and ?
PS. Also, when and are functions of , does there always exist ?
I might mistake, but it doesn't seem right to me. What does mean ? If it's the differential of in , then it's a linear application and your equality isn't homogeneous.
by a simple composition of and .
If , then we express it like that :
@courteous : can you precise your goal, because your function is not of the form .
Hello pece! I want to differentiate the slope/angle of a parameterized curve with regards to parameter ... that is, I'd like to find at a point .
EDIT: And so that everything is summed-up in this one post, this is what I've figured out so far (not necessarily correct though):
Now, I don't know what to do with the term.