Page 18 - Calc for the Clueless

Definition

Suppose y = f(x). The derivative of f(x), at a point x denoted b, f'(x), or dy/dx is defined as

if it exists.
Note 1

All mathematics originally came from a picture. The idea of derivative came from the slope. Now the definition
is independent of the picture.

Note 2

If y = f(t) is a distance as a function of time t

is the velocity y(t).

Well, heck. Note 2 about velocity is not enough! Let's do some examples.

Example 1—

Suppose y = f(t) stands for the distance at some point in time t. Then f(t + ∆t) stands for your location later, if ∆t
is positive. (Remember, ∆t means a change in time.) y = f(t + ∆t) - f(t) is the distance traveled in time ∆t.

If we take the limit as ∆t goes to 0, that is,

then f'(t) is the instantaneous velocity at any time t.

Note 1

The average velocity is very similar to the rate you learned in elementary algebra. If you took the distance
traveled and divided it by the time, you would get the rate. The only difference is that in algebra, the average
velocity was always the same.

Note 2

Even if you drive a car at 30 mph, at any instant you might be going a little faster or slower. This is the
instantaneous velocity.

Example 2

Let f(t) = t + 5t, f(t) in feet, t in seconds.
2
A. Find the distance traveled between the third and fifth seconds.