Intro

We cover what a derivative and integral is, how (and why) they are functions themselves, their relationship, and the intutive meaning of the Fundamental Theorem of Calculus.

Derivatives

A derivative is the rate of change of a function. You probably know slope as "rise over run" or the "change in y divided by the change in x". Let's think more about these definitions.

In our first function analogy (using a display panel), the derivative is how much the number changes if you take a step forward. This is the change in y (function) divided by change in x (position). We are letting the change in x be 1, so the derivative is simply the change in y divided by 1, or the change in y.

In the second analogy, the derivative is the steepness of the hill. It is the change in height for a horizontal step forward.

Observations

1. The derivative of a function is itself a function. Recall that a function is a maps (associates) one number to another. The derivative takes a number x (your current position), and tells you how much f(x) will change if you take a step forward. The derivative is how much the function is changing at that point. Think of the derivative as guiding the function.

2. The derivative can predict the function in the short term. At any point we know how much the function is changing. Thus, to find the next point, we can simply take our current point and add the change. You can multiply the derivative by how far you are going to roughly predicit the new value.

For example, if you have $100 (current value of function) and are currently earning $200 a year (your derivative or rate of change), in 1 year you will have $100 + $200 * 1 = $300. In two years you will have $500.
Intuition:

New function value = current value +  (how much function is changing per step * how far forward you go)
                            OR
              New value = current value + derivative (at current point) * how far forward you go
             
Math: (linear approxomation):        f(x + a) = f(a)+ x* f'(a)

a = our current position
x = our change in position

Note: this is only an approximation. If the derivative changes (discussed later) then this will only be approximate. This is a linear approximation because our approximation can be graphed as a line.

2. The sign of the derivative tells you how to improve the value of your function. If the derivative is positive, it means your function will increase if you move forward. If your derivative is negative, it means your function decreases if you go forward. To maximize the value of your function, move according to the derivative. If the derivative is zero, it means your function is not changing. You are either at the maximum or minimum.

3. The size (magnitude) of the derivative tells you how much you can improve your function. A small derivative indicates a gradual change, and thus a gradual improvement (and vice versa for a large derivative). Small and large are relative to the value of the function (i.e. a derivative of 100 is large if the function's value is 25).

4.  The derivative controls the function. As in point #1, the tangent line points to where the function is going to go next. The next point is simply the current point + the derivative at that point. If the derivative becomes zero, then the function does not change. The derivative tells the function whether to get bigger or smaller, and by how much.

Derivative of the derivative?

If the derivative controls the function, what controls the derivative? Good question! Let's think about this. A function is a mapping of one number to another. The original function maps the x value (some location) to another number y, according to some rule. The derivative maps the x value to how much the function will change if you go forward by 1 step.

The derivative does not know the value of the function -- it only knows the change in the function.

The second derivative is the derivative of the derivative. It controls whether the first derivative increases or decreases. What does this mean? Well, the first derivative controls the rate of increase/decrease of the first function. The second derivative controls whether this rate will increase, decrease, or stay the same. The second derivative controls the first derivative, and the first derivative controls the function.  Let's look at a typical conversation.

Function (equals 0):  Hey, how was your weekend?
First deriv (equals 5):  Increase by 5!!
Second deriv (equals 3):  Increase by 3 yourself, first derivative!!

Function (equals 5):  I like cotton candy.
First deriv (equals 8):  Increase by 8!!
Second deriv (equals 3):  First derivative, keep increasing by 3!!

Function( equals 13 ):  Man, I love improving my value.
First deriv (equals 11):  Increase by 11!!
Second deriv (equals 3):  First derivative, keep increasing by 3!!

Function (equals 24 ):  I have 3 toes.
First deriv (equals 14): Increase by 14!!
Second deriv (equals 3): First derivative, keep increasing by 3!!

Observations
1. Each function directly controls the one above it. However, functions do not know the starting value of other functions. All they know is how much to change them.

2. The "biggest" derivative indirectly controls the function (and whether it goes to infinity or negative infinity). The second derivative was the greatest derivative in our case. Because it was positive (it said to keep increasing by 3), the function must eventually be positive. Even if the first derivative said to decrase, the second derivative would ultimately make the first derivative positive. After that, the function would start to become positive. In this function, we know that as we let time go on, it will eventually go toward infinity. If the second derivative was negative, the function would go towards negative infinity as time goes on.