A good teacher of mine defined Calculus as the study of change.
I see Calculus as a series of cool tricks to make particular calculations much easier than they would otherwise be. I'll cover where the first tricks you would likely learn in a Calculus class come from.
Let's look at this graph of y = x².
Slope at a point
To better understand what's going on, we'll rewrite the equation into a function f(x) = x². F(x) being a combination of a function declaration "f()" and the variable to be plugged in "x."
Now if we want to find out the exact slope of the function at x=1, how do we do it? One way could be to zoom in indefinitely into the curve at x=1. That way it will start to look like a line, and we could fairly accurately say what the slope is. But that’s like looking at a square drawn on a piece of paper and saying yeah, its corners are all 90 degrees. But no matter how hard you try you can never hand-draw a perfect square. It will always be slightly off. Even if by the tiniest amount of a micrometer.
So we need to precisely measure the slope of the function. We can use the slope formula (y₂-y₁)/(x₂-x₁) to get a good guess of the slope. If we take, say x₂=1.1 and x₁=1 and we bring x₂ closer and closer to 1 without it ever actually reaching 1. We can get quite an accurate estimate of the slope at x=1.
I put in smaller and smaller values in the table on the left which mean the difference between x₁ and x₂, and as you can see the values on the right that show the resulting slope get closer and closer to what seems to be 2.
It is tempting to plug in 0 to the g(x) function above which calculates the slope for us. The only problem is, if we do, it will result in a division by zero.
The Calculus slope formula
First, we should understand how that function works. It stems from the basic slope formula.
We replace the y’s on top with the function where we plug in the x’s.
Then, based on the fact that x₂ is really just x₁ + something, we replace it with that and we call the “something” h.
And the last step is to simplify the resulting equation by removing the x₁’s on the bottom that cancel each other out.
Derivatives in Calculus
Now let’s plug-in the function into the equation and see if we can get rid of the h on the bottom.
First step is to open up all the parentheses.
Next the two x² on in the numerator cancel each other out
In the following step the h’s cancel each other as well and we’re left with 2x+h.
Now we can finally plug in 0 in h, and we get 2x.
If we plug-in 1 into the x now, we will indeed get 2. The slope we guessed with the table on desmos earlier.
Now let’s look back at the graph, there’s something interesting about the function 2x that we got.
Notice, if we plug in any x value into our function, we will get the slope of the initial function x² at that x value. 2*(.5)=1, 2*(2)=4, and each of them turn out to be the slope at the x value we plugged in. That means our equation 2x is an equation we can use to get the slope of our initial function at any point. In other words, a derivative. A derivative of a function f is a function f’ that shows the slope of the function f at any x value, given that the function f is continuous. The reason for the continuity of the function is that if it’s not continuous, the result of the derivative will be undefined at the discontinuity of the function the derivative is taken from.
Now that we know all this, we can understand something called differentiation rules, and appreciate how much they make life easier for us.
For example, the differentiation rule we use for x² is called the power rule:
We’ve learned how to precisely calculate the slope of a specific point on a curve, we’ve learned about derivatives. These are some of the first things you’ll learn in Calculus classes.
I hope you enjoyed reading this article. Make sure check out my other posts, and I'll see you later 🙂