## The Derivative at a Point |
HELP |

We denote this slope by `f'(c)`, and we say `f'(c)` is the derivative of `f` at `x = c`.

[`f'(c) = lim_(h->0) (f(c + h) - f(c))/h`]

With the applet above, we'll explore where this formula comes from. Note that
in the applet you can move two points on the
`x`-axis -- one with `x`-value `c`, and one with `x`-value `c + h`.
- Start with `c = 2.5` and see how `h` changes as you move the other point. When is `h` positive? negative?
- Show the tangent line and the secant line, and see how the secant line approximates the tangent line when `h` is very small. What happens when `h = 0?` Why?
- How is the fraction `(f(c + h) - f(c))/h` related to the formula for the slope of a line? Fill in the blanks: `(f(c + h) - f(c))/h` is the slope of the line between points ______ and ______.
- How are `(f(c + h) - f(c))/h` and `lim_(h->0) (f(c + h) - f(c))/h` different? In other words, what is the effect of sticking that limit out in front of the fraction?
- Hide the secant line. Notice that the graph is "pointy" at `x = 1`. What happens to the tangent line when `c = 1`?
- Show the secant line again, and keep `c` at `1`. Does `lim_(h->0) (f(1 + h) - f(1))/h` exist?
- Confirm by playing with the applet: `f` is
*continuous*at `x = 1`, but `f` is not*differentiable*at `x = 1`.