This is a simple question in differentiation.
Let for any real number , determine at (1,-1).
Some students may give the following:
– – – – – – (*)
Then differentiate both sides with respect to , yield
Put , yield
The correct answer should be .
The main problem appears in (*).
is true only when .
If , we will have the following
– – – – – – (**)
As I mentioned in “16 踢 #2″, we should pay attention when taking square roots (say), that is
is NOT equal to ; must be non-negative, hence we should write , that is,
Back to the original question, we would like to find the first derivative at (1,-1), and the -coordinate is negative. The fact is
and we should find the derivative out of the following relation
And the correct answer will be figured out.
1. May be, it is better to start with , try.
2. How about when students start with
3. Let ( is an even positive integer) for any real number , determine at (1,-1). [You may consider cases of and .]
Urm, here is something about ““, just read the old post.