Hello, mathaction!
I must assume you know the derivative of the arctangent.
. .
We have: .
Chain Rule: .
We have: .
Therefore: .
This one has a surprising answer . . .
We have: .
. .
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
How come it works out to zero?
Let
Then: .
is in a right triangle with:
Here is the triangle: Code:
*
* *
* *
* * x
* *
* α *
* * * * * * * * * *
1
Let
Then: .
is in a right triangle with:
Here is the triangle: Code:
*
* β*
* *
* * x
* *
* *
* * * * * * * * * *
1
Get it?
and are in the same right triangle.
They are complementary: .
We just differentiated a constant function: .