Learn Calculus in Five Minutes!*
*Got your attention? There is a slight caveat to my title of 'Learn Calculus in Five Minutes', that caveat is I'm going to show you how to derive a simple equation. You're not going to be replacing Liebniz (or Newton) in the math books after this article, but you will have a much better understanding of what calculus is and what it does.
Calculus 1 - The Derivative
The derivative can be interpreted geometrically as the slope of a curve, and physically as a rate of change. Because derivatives can be used to represent everything from fluctuations in interest rates to the rates at which fish populations vary and gas molecules move, they have applications throughout most of the sciences and real world.
The slope of a curve on a graph is representing the change between an infinite amount of individual points in time, measuring it's change over that time set. Each one of those points will tell you the exact velocity of that object at that precise time. If the graph is accurate enough, you can draw a vertical line from the X axis up to where it intersects the curve and then draw second horizontal line at a 90 degree angle back to the Y access.
This will give you the X and Y coordinates for that specific point. However, what if you wanted every possible X and Y coordinate for a function? It would take forever, literally forever, to draw and redraw lines and recording the coordinates, as there are an infinite number of points. There must be an easier way to find the coordinates!
If you were to compare two points on a graph, you'd be able to see the rate of change between the two points. This will tell you the average velocity of the function over that defined time period. However, with an infinite number of coordinates, and an infinite number of combination of coordinates, it would take near infinity to find all of the answers along the slope of a single equation. Calculus makes this much easier.
Calculus uses all sorts of fancy algebra and trigonometry to explain why what is going on. I don't want to get bogged down in limits and difference quotients. Instead, you'll need to trust that they math actually works and has been working since 1684 (the year Liebniz is credited of first publishing about calculus).
The Power Rule
This is by far the easiest rule when deriving in calculus. Powers of X are then also the easiest to derive. The simple math behind the power rule is:
d/dx (xn) = nxn-1
So, a very easy example would be:
x2 which derives to 2x.
x3 derives to 3x2.
x4 derives to 4x3.
See, what we're doing there? We're taking the power of the X and moving it down and multiplying it with the the base, while subtracting one from the power. In the examples above the base was always 1. This holds true for any real constant power (even for negative bases or powers!) Below is an example of using the power rule on equations that don't have a base of 1.
5x4 derives to 20x3
-3x2 derives to -6x
2x-3 derives to -6x-4 (remember we're ADDING to the power)
Now what happens if I don't have a power to the base, but rather just a number. Think about what a number looks like on a graph. If you have an equation of y = 3, the graph would be a straight line horizontally across forever (graph shown above). So no matter where you take a derivative on that line, the velocity of the equation will always be 0.
3 derived is 0.
Now that we understand what a derivative is, can we do it more than once? Yes! What we've been doing so far is called the first derivative. As demonstrated, eventually all functions will be derived until they become 0. However, as an example I'll show how that happens. We'll start with the function below:
x3 derives to 3x2 this is the first derivative that tells us velocity of the function
3x2 derives to 6x this is the second derivative and tells us acceleration of the function
6x derives to 6 this is the third derivative and tells us the jerk or jolt of the function
6 derives to 0 this is the fourth derivative and is tells us the jounce of the function
The Power Rule and Polynomials
What happens when we have more than just a power of x? It's much the same. Examples are shown below:
5x2+ 3x +2 derives to 10x+3
3x4- 2x3 + x2 - 100 derives to 12x3- 6x2- 2x
Now these are very basic examples and I don't expect you to start planning rocket launches to the moon for NASA with your new found understanding of calculus, but it's the building block for all infinitesimal single variable calculus. There are many, many other rules and exceptions when you begin adding in more than single bases with powers and addition and subtraction. These are the beginning to a greater understanding of the calculus.