Derivative

derivative

Let $f : I \to R$ where $I$ is an interval (open, closed, half-open/closed), which is definitely connected. Then $f$ is differentiable at $c \in I$ if:

lim_{x \to c} \frac{f (x) - f (c)}{x - c}

exists. (see Functional Limit for details, and highlighting Existence of Functional Limits for showing this).

Note that:

If $f$ is differentiable at $c$ , then we define $f^{'} (c)$ to refer the value of the limit:

lim_{x \to c} \frac{f (x) - f (c)}{x - c} = f^{'} (c)

specifically we call this the derivative of $f$ at $c$ .
2. Similar to One-Sided Limits, if $c \in I$ is an endpoint of $I$ (ex: a closed interval), then this limit coincides with a one-sided limit (which we refer as a one-sided derivative):

lim_{x \to c^{-}} \frac{f (x) - f (c)}{x - c}, lim_{x \to c^{+}} \frac{f (x) - f (c)}{x - c}

If $f$ is differentiable at every $c \in I$ then we say $f$ is differentiable on $I$ .
$\frac{f (x) - f (c)}{x - c}$ is the slope of the secant line. Taking its limit gets the tangent line at a point.