The slope of the function f at the point (a,f(a)) is the limit of the slope of the secant line between x = a and x = a + h as h gets closer to 0. Whew, that's a mouthful.
From the formula for the slope of this secant line, we find another formula:
"The slope of f at a" is also called the derivative of f at a and is written f'(a). This nice formula is known as the limit definition of the derivative, and we write it again here, with correct notation:
Visually, we move h closer and closer to 0 (equivalently, move a + h closer and closer to a) and watch what happens to the secant line, as in this animation.
If we could keep going until h reached 0 (equivalently, until a + h reached a) we would find a line that, instead of passing through the graph twice, would hit the graph at one spot and bounce off, as shown in this picture:
The slope of the function f at the single point x = a is the slope of this line, also called the tangent line.
Two important things to remember:
- The slope of f at a and the derivative of f at a are the same thing.
- Since the derivative of f at a is a limit, the derivative won't always exist
Another phrase for f'(a) is the instantaneous rate of change of f with respect to x when x = a.