Foundation for finding derivatives of function?

Tom2013

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Jul 17, 2013
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My question is: What aspects of functions are critical as a foundation for finding derivatives?
 
What aspects of functions are critical as a foundation for finding derivatives?
Which definitions and theorems have you reviewed? What "if...then" statements have you found? What are your thoughts thus far?

Please be complete, so we can "see" where you are needing help. Thank you! ;)
 
My question is: What aspects of functions are critical as a foundation for finding derivatives?

Piggybacking on what Stapel said, this is a pretty broad question. Was there anything in particular that you were doing that sparked this question from you?
 
In particular, what do you mean by "aspects of functions"? The derivative of a function f(x), at x= a, can be defined as \(\displaystyle \lim_{h\to 0}\frac{f(a+h)- f(a)}{h}\), if that limit exists. If it does not then the derivative does not exist. In particular, since the denominator obviously goes to 0, the numerator must also go to 0. From that it follows that f must be continuous at x= a. But that is not sufficient. The function f(x)= |x| is continuous at x= 0 but not differentiable there.
 
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