First I would like to mention that and are the same! Because the function is continous on some open disk.
(But since you are in mechanical engineering you probably do not care about that).
Thus,
Now take the derivative along y on the second one,
And finally,
Question 1 a) I presume.
i)
First note that . The rest is an exercise in the chain rule:
So
================================================== ======
ii)
================================================== ======
iii)
First, this needs to be done by implicit differentiation.
Second: .
So:
This needs to be neated up a bit:
I'll leave it like this.
================================================== ======
iv) Ah, a practically straightforward problem!
Using the quotient rule:
-Dan