Something I do recall from high school calculus class is that the derivative of is . This is one of those little nuggets of information that really helps you through bigger, tougher equations. But why is this so? Down the rabbit whole we go.

First, we need to define what a derivative is (again). Its just the slope of a line at a given point. Slope is just rise over run, or the change in y or the change in x: or . In simple geometric terms this is pretty easy. Let’s go back to our simple line:

Using just a paper and pencil we can figure out the slope of this line at any given point. We can probably come up with the slope of any line at any other given point, too. In this example finding the first derivative, and its meaning, are very easy. But how do we do it with tougher examples? Well, lets look at the idea of finding the rate of chang of x at a given point. This is tough to do. We really need two points to find the slope. How do we pick another point? On a tight curve, using our point at x and another point will probably give us a false (but perhaps close) slope. What we need is another point that is infinitely close to x. Let’s say we have x and another value, x + h. Our equation to figure out the slope based on these two values would look like:

This is pretty straight forward so far. Our value h needs to be very, very small to give us an accurate slope. In fact, to give us a precise slope it has to be approaching zero. So what if we create a limit where the difference in our points is disappearing?

This is pretty and all, but it doesn’t really get us anywhere. Remember, we are trying to find some general rules to find derivatives. Here we need to take another “guess” at trying to find an equation we can simplify. Our guess is going to be to use as a starting point. We do this because we want do be able create multiple terms and hopefully find a way to get rid as much junk as possible.

#1 – Starting equation:

#2 – Plug into our slope formula:

#3 – Expand:

#4 – Multiply:

#5 – Add (or subtract):

#6 Divide by h:

So we have a general formula that if the . For our dead-simple line , the derivative is or 1.

If you recall, we started with the derivative of sin x being cos x. So let’s plug and chug. It’s calculus, but it looks like algebra!

#1 – Starting equation:

#2 – Slope equation with sin plugged in:

#3 – Trig: , which simplifies to . I had to look all of this up, which is why I titled this post the rabbit hole.

#4 – Pull h/2 over to sin:

#5 – Take the limit as h approaches 0. The limit of a product is the product of the limits. All of the h/2 elements go to 0, basically leaving cos(x).

To figure this out I had to relearn some trigonometry, the basic theorem of derivatives, and the limit product rule. Math is hard; let’s go shopping!

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