Problem: Derive (3) and (4) from (2).
First, let's solve for (3).
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: