The limit
is called the derivative of f(x) at x. You can think of this limiting process as creating a completely new function called f-prime of x from the original function f(x).
We saw last time that the tangent line to the graph of f(x) at a point (a , f(a)) is a line whose slope is the limit of slopes of secant lines. The secant lines pass through points (a , f(a)) and (b , f(b)) on the curve. In the limit as b moves close to a, the slopes of the secant lines approach a limiting value that is the slope of the tangent line.
Here is how the derivative and the slope of the tangent line are connected. If we substitute a for x in the definition of the derivative, we get
The slope of the tangent is the limit of slopes of secant lines:
To see that these two things are the same thing, simply write b = a + h and note that saying that b approaches a is the same as saying that h approaches 0.
Thus,
Last time we saw that an instantaneous rate of change is a limit of average rates of change as the interval shrinks to 0. In the language we were using last time,
To see that this is just the derivative, write x2 as x1 + h:
From this we see that
Since the only thing we have to go on at this point is a definition, computing derivatives can be potentially very difficult. Here are some examples.
Section 2.8: 7, 8, 13, 18, 27
Section 2.9: 7, 8, 30