So, my homework problem is deriving -2*sin(x)*cos(x) as part of using the concavity theorem on -6x^2+cos^2(x). My calculator is saying 2-4(cos(x))^2, but I'm not getting to the same answer.
My other question is working the problem sin^2(cos(4x)), this was a problem on my exam that I got right, but only because I memorized the general pattern of the problem when we got it in class the week before and she just happened to have it on the test. I need to see it worked out step by step, as I get lost in it pretty quickly.
( It works out to -8*sin(4x)*sin(cos(4x)) * cos(cos(4x)) )
Thanks!
For your 2nd question (which hasn't been answered):
Use:
The Chain Rule
If this is the "general pattern" you memorized, you did the right thing.
In this case you want to differentiate
Using the chain rule:
Note that f(x) is itself a combo function with the trapped inside the
So the derivative of is
A trig identity says that
So
Notice the g(x) is also a combo with the 4x trapped inside the .
So the derivative of is
Now using the master chain rule:
The derivative of is
Substitute g'(x) and substitute g(x) for x in the f'(x) calculation:
Phew! And that's the answer.
NOTE: This answer is a bit more simplified that the one you got because I used the trig identity . You can check that they are both the same function, by graphing them or randomly checking points such as x=3. I have checked and saw that they are the same, so my answer is also correct.
Hope that helps
Mathemagister