Derivatives From Top To Bottom

There is a lot of unreasoned stress around calculus and how difficult it is, however the idea of derivatives is really simple.

Before diving deep it is really important to have a fundemental understanding of what a line is on a graph. Sounds like a very shallow topic, after all its just a line!?

Functions On Graphs

Every line represents a function. It is easy to imagine a function as a machine that takes in a number and spits out a new one based off of that number. How does it spit out that number? Well, that depends on the insides of the machine. f(x) is a function. Function f takes in a number x, and puts out another number y. Here are some examples:

f(x) = 2x

This function takes in a value x, doubles it (multiply by 2) and puts out that number.

f(x)=x²

This function takes in a value x, squares it and puts out that number squared.

f(x) = √(x) + 1

This function takes in a number, takes that number’s square root, adds one and puts that out.

In every single function you can think of, there are 2 numbers involved. The input (traditionally called x), and the output (traditionally called y). If this all makes sense, then you will understand how functions are reflected to graphs.

What we do is take the input and output, and plot it on the coordinate plane. We do this by going right the input amount of steps. If the input was 2, then we would go right 2 steps. If the input is -3 then we go right -3 steps which is basically taking 3 steps to the left.

As we can see the the blue dot went to the right 8 times, because the input was 8. the dark green dot went to the right 4 times because the input was 4. The lighter green dot went to the right 0 times (or stayed where it is) because the input is 0. You get the flow. For this reason, we label the horizontal line x. x means input in graph mathematics.

So lets imagine we have a function f(x) = 0.5x. We take in each of the colorful dots and put them through this function which basically halves the number put in.

So as seen in the figure above, if we put the blue dot and the red dot in the function, the functions return the half of the input giving us how many steps we should go up. we put 8 through function f(x), we get 4 so we go 4 steps up with the blue dot.

This is the same for the red dot. In f() we input -12, which returns -6. For the red dot we go up -6 steps, or in other words go down 6 steps.

The coordinates of the blue dot are (8, 4) or in other words, (8, f(8))

The coordinates of the red dot are (-12, -6) or in other words, (-12, f(-12))

So if you are used to graphing in the x-y coordinate system, keep in mind that we are doing the exact same thing here but y is replaced with f(x)’s output.

Problem For Understanding

Just to make sure we are all comfortable with this, here is a problem to think of:

If we are given the x-coordinate (input) and f(x), how do we find how many steps the point went up/down?

This is actually pretty simple, how many steps a point went up or down is dependent of the input via f(x), so the “altitude” of the point is the output of the input put in the function. So the coordinates of that point would be (x, f(x)).

Therefore to find how much the point went up/dpwn we have to just process the input thought the function.

Dots Make Lines

Plotting points must be easy, but how do functions make lines? Well, you have come to the right place my friend.

This was mindblowing to me the first time I learned it.

Lets take the function f(x) = 0.5x, and plot 2 points randomly. This doest really matter at all what we pick so lets just go for 10 and -10.

Now that we plotted these 2 points, lets plot many many more.

So I plotted a bunch of dots, but it still is not all the points, I can fit even more points in between 2 points.

We can plot an infinite amount of points. And guess what happens when an infinite amount of points come together? We get a line!

Ladies, and gentelmen, I present to you, the graph of f(x) = 0.5x !!!!

Slope

If everything made sense above, now we are ready to move on to the real stuff!

If you have ever heard the definition of slope, they talk about the slope of a line, but completely diregard that perspective. The slope can be taken between any 2 pairs of points. It is nothing more than the difference of the y-coordinates divided by the difference of the x-coordinates of any 2 points.

That last paragraph might have been confusing but here is the definition of slope you should consider: The slope is rise over run of a straight line between 2 chosen points. It is basically how steep it is!

So lets use the points above for an example.

Δx means distance from x1 to x2. Or in other words x2-x1

Δf(x) means distance from f(x1) to f(x2). Or in other words f(x2)-f(x1)

Now the slope is Δf(x)/Δx, so if we expand that we get (f(x2)-f(x1))/(x2-x1) (Green Text in Figure 8)

So the slope of the orange line (when we give the points real values) turns out to be 0.5 as seen.

One important thing to know is any 2 points picked on this line segment have a slope of 0.5 between them since it is a straight line.

My Recommendations on the Next Section

If you are not rock solid with the last 1008 words of this article, this part wont make too much sense. I cannot emphasize enough how important going back and understanding it is, but i guess I can’t stop you.

Derivatives

Ah, you are here finally.

First lets talk about the slope of a curved line.

What if I asked you to find the slope of the curve y = 3x² at x=0.2? In math that is asking what the slope of the tangent line at x=0.2 to y=3x² is. Here is what a tangent at x=0.2 looks like.

The orange line is the tangent to the red line at point x=0.2.

A tangent is a line that skims another line at only one point. In this case the tangent is the orange line that only touches the red line at x=0.2

But how do we find the slope of that orange line?

To find that we plug in x to the derivative of function f (in this case 3x²).

The derivative of a function describes how the slope of the tangent line changes as its touching point’s x coordinate changes.

The derivative of 3x² is 6x. I will get into how that is dervied further in the article.

If we want to find the slope of the tangent line at x=0.2 we substitute 0.2 instead of x in the expression 6x. So now we know that the orange line’s slope is 6 * 0.2 = 1.2.

If we wanted to know the slope of the tangent line at x=5, for example, we would substitue 5 instead of x in the expression 6x and we would get 30. The slope of the tangent line at x=5 is 30.

Since using this derivative idea we can tell the slope of any line we call the slope of a curved line it’s derivative. Thats just tradition.

And that’s really all there is for derivatives, you got em’ now!

Thats All For Today, Folks!

It wasn’t too difficult after all, right?

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