Calculus Is Overrated – It is Just Basic Math

let us talk about calculus I guess you all are familiar with the equation of a line we all have studied it in our school days right we write it something like y equal x where if we have this line then at every point on the line the value of y depends on the value of x for example if x equal 2 then y equal 2 right this relationship shows how y changes as X changes but this kind of relationship is easy to visualize because it's a straight line constant and predictable now imagine you're a curious thinker in the 1600s when the world was just beginning to explore the motion of planets the falling of objects and how things move and change over time straight lines didn't cut it anymore things in nature didn't always behave predictably or move in straight lines think of a ball rolling down a hill or water flowing into a basin how do we even begin to describe such motion mathematically let's go back to the straight line for a moment a straight line has a very special property it has a constant slope the slope tells us how steep the line is that is how much y changes when X changes for y equal x the slope is is one meaning for everyone step forward in X Y goes up by one right now consider another line Y = 2 * X the slope of this line is 2 meaning that for every one step forward in X Y now goes up by two in other words this line is much steeper than the first one therefore slope is like a measure of the line's steepness it tells us how fast Y is changing compared to X now imagine a situation where the slope is zero for instance if you have a line like y = 3 a horizontal line then we know that the slope of this line is zero because y doesn't change at all no matter how much X changes on the other hand if the slope is a very large number the line is super steep almost vertical I think you understood my point but then someone like Galileo might come along and say what about something like y equals x^2 that's not a straight line it's a curve for example we all know that distance equals speed into time now assume you are running at a constant speed say 2 m/s so D will now be equal to 2T where D is the distance and T is the time right now if I draw this graph where I put time on X AIS and distance on yis then we get this straight line and the slope of this line will be the same as your constant speed of two but now assume you start running in a race and instead of maintaining a constant speed you begin slowly and then gradually speed up at first you might cover only 1 meter in the first second which is equal to this point then 3 m in the 2 second and perhaps 6 m in the 3 second the distance you cover is no longer proportional to time and the graph of distance versus time will no longer be a straight line it will be a curve in this case your speed or the slope of the graph or the steepness of the curve keeps changing at the start your speed or the slope is small because you're running slowly as you speed up the the slope becomes steeper showing that the distance you cover in each second is increasing therefore at any given moment the slope of the curve represents your speed at that exact point in time back in the 1600s this was a huge problem the slope wasn't constant anymore people could draw the curve but they had no idea how to calculate the slope at any specific point on it and then came Geniuses like Isaac Newton and gotfried vilhelm libbets in the late 1600s both of them working separately came up with a groundbreaking idea instead of thinking about the whole curve at once they zoomed in really really close so close that the curve looked almost like a straight line this is how the concept of the derivative was born we all know that if we have two points like this on a straight line then its slope is given as y 2 - y 1 /x 2 - X1 Newton and lietz realized that to find the slope of a curve you could look at two points getting closer and closer together such that they could find the exact slope at a single point but how does it work let us understand this with an example suppose we have a curve y = x^2 now if we want to find the slope of this curve at say X equal 1 then assume we take two points where the first point is obviously one and one square or one itself and then let us take another very close Point say x = 1. 1 and Y = 1. 1 square or 1.

21 now these two points are so close to each other that the curve here behaves almost as a straight line so what will be the value of this slope it will be this minus this over 1. 1 -1 right this gives the value of slope at this point as 2. 1 this is simply amazing now let us level up our gear a bit more assume we have a point as X and the Y value as x square and another point at some distance H from X say x + H such that its y value will be x + h s so now what will be the slope of this line it will be x + h 2 - x^ 2 / x + H - x look at the denominator X will get canceled out and we are only left with h now expand the numerator to get this as x² + 2 x * H + h² and minus this x² oh look now x² will get canceled out and we are left with 2x * H + h² over H divide it properly to remove H from the denominator and we are left with 2x plus h here comes the magic we want both these points to be as close as possible which means this H is so so small that it is almost invisible to our naked eye thus we say that H tends to zero and since H is not exactly zero we put this liim here which stands for limit which means what is the value of this thing when H is very close to zero and to solve for it we put H as zero which gives us 2x as the slope of the curve y = x² so we say that the derivative of y which is represented using dy/ DX which indicates the change in y with respect to X at a given point is 2x when y = x^2 now at x equal 1 what will be the derivative of y with respect to X or in other words what will be the slope of this curve at x equal 1 just substitute X as one here and we are done the slope of this curve at x equal 1 is 2 right we solved it previously and got our answer as 2.

1 remember which was so close to two but not exactly two because the value of H was 0.