## The Derivative as a Function

HELP

We know that if `f` is a function, then for an `x`-value `c`:
• `f'(c)` is the derivative of `f` at `x = c`.
• `f'(c)` is slope of the line tangent to the `f`-graph at `x = c`.
• `f'(c)` is the instantaneous rate of change of `f` at `x = c`.
In this applet we move from thinking about the derivative of `f` at a point, to thinking about the derivative function.

#### Explore

1. Move the blue point on the `x`-axis to change the value of `c`, and observe how the line tangent to the `f`-graph at `x = c` changes. When is the slope of the tangent line greatest? lowest?
2. Can you find (at least) one value for `c` where the tangent line crosses the `f`-graph only once, at its point of tangency?
3. Check the box to see the point `(c, f'(c))`. Do you see how the slope of the tangent line is being plotted?
4. Check the box to trace the red point. Now drag the blue point around. The resulting red graph is the graph of the derivative function. That is, it is the graph of the function `f'`.
The derivative of `f` at the point `x = c`: [`f'(c) = lim_(h->0) (f(c + h) - f(c))/h`] The function `f'`: [`f'(x) = lim_(h->0) (f(x + h) - f(x))/h`]