Problem: Derive (3) and (4) from (2).
Given:
(2)
(3)
(4)
Solution:
First, let's solve for (3).
Recall that:
,
and
.
We can use this information to replace the differentiating terms accordingly.
After doing so, we get:
but knowing that
, we can rearrange the terms to get
.
Using this information in the previously derived equation, we find that:
.
Finally, after differentiating
with respect to
, we get:
.
Now, let's solve for (4).
Once again considering that
, we can solve for (4) by differentiating
twice and then plugging it into (2).
Deriving twice, we find that:
After plugging this into (2), we see that:
.
Once we rearrange
, we find that
.
We can plug this in to the above equation to get the solution:
.