How do you derive the formula for the derivative when C is known and when it is not? I know that the formula for the slope of the tangent line when c is known is and that the slope of the secant line is  but I don't really understand how. When c is not known, I don't understand what h is and how that formula is derived:
 A: Suppose you have a point $c$, as you've noted the slope of the tangent line to $f$ at point $c$ is given by: $$f'(c)=\lim_{x\to c} {f(x) -f(c) \over x -c}$$
We can introduce a new variable, $h$, that's dependant on $x$ through:
$$h = x-c$$
Notice that we can rewrite $x$ in terms of $h$ by:
$$x = h + c$$
Thus allowing us to change our dependence on $x$ to a dependence on $h$, which allows us to rewrite our first known as:
$$f'(c)= \lim_{h+c\to c} {f(h+c) -f(c) \over h+c-c}$$
Notice that when $h+c\to c$, that is the sum $h+c$ approaches $c$, you have $h\to0$, that is $h$ approaches $0$, and vice-versa. Thus allowing us to rewrite this again as:
$$f'(c) = \lim_{h\to 0} {f(c+h) - f(c) \over h}$$
This formula is independent of $c$! and so, you can put anything regardless if it has known value or not, and so by simply changing the $c$ to be an $x$ to aid visualization that the choice of the point where you calculate the slope is arbitrary you get:
$$f'(x) = \lim_{h\to0}{f(x+h) - f(x) \over h}$$
This is how you'd derive this formula, I'll be sure to update / reply in a comment if need be.
