How do you find f'(x) using the definition of a derivative for #f(x)=cos x#?
2 Answers
See the explanation.
Explanation:
When
since
The answer is
Explanation:
This is the definition of the derivative:
So we know that
Now, before we progress any further, we need to lay down two facts:

#lim_(x>h) (cos(h)1)/h = 0# 
#lim_(x>h) sin(h)/h = 1#
If you'd like to verify these yourself, you can just plug these functions into your calculator, and see what they approach as
Now, if you noticed,
So, let's use this to split up
Now we can factor out a
And if we did some algebra:
And applied our first identity:
And now if we factored out a
Now we can apply our second identity:
And since there are no
Of course, this is the ultralong way to derive the answer to this, and most calculus teachers just prefer you simply memorise this. And practically speaking, it's easier than deriving it all over each time.
The first 11 minutes of this video do this too. The only difference is that this video derives
Hope that helps :)