So let's say you're on a physics class and you're solving a sum. You have to find the speed of a car that moves in a way shown by this distance versus time graph. Well, that's simple, right?
You can just use the formula speed is equal to distance upon time, which will be the gradient of this graph. That's rise over run. And you get the answer.
But in the real world, the same car is much more likely to move like this or this. So what would you do if you had to calculate the car's speed at some time of t seconds now? The gradient of the graph and thus the speed is constantly changing at every instant in time. So your answer would be very hard to pinpoint.
Let's try doing the same thing earlier by drawing a line and finding its slope. This line intersects the graph at the new point t1. The gradient of this line would be the rise which is equal to f of t1 minus f of t over the run which is t1 minus t. Let's take the difference between t1 and t to be delta t to simplify the equation a bit. So now the equation will become f of t plus delta t minus f of t over delta t.
Evidently this line is an inaccurate depiction of the speed because it's far from what the actual curve represents. When the value of delta t gets smaller and smaller, this line becomes a better and better approximation of the gradient at t. Another way to think of this is by taking smaller values of delta t, you're really looking closely at the point t. And as you notice, over really small intervals of time, the so-called curve actually looks like a straight line. So we can write that the line represents the car speed better and better.
as delta t approaches zero, that is it becomes smaller and smaller. There's a special symbol in calculus called the limit to represent exactly what we said, that is delta t gets smaller and smaller. Now we have the final formula for the car speed at t. Now let's connect this idea to a sum to make it clear. We have to find the gradient of the equation fx is equal to x squared at some point x.
We already found an equation for the gradient of that graph, but it applies here too. So we'll just replace delta t with delta x and t with x. We know this is an x squared graph, so we can square both of these terms. Let's expand x plus delta x the whole squared using simple algebra. Notice that the x squared terms will cancel.
Now we can take delta x common and it will cancel in the delta x below finally giving us the limit as delta x approaches zero of 2x plus delta x now remember to get an accurate answer delta x is supposed to approach zero and become infinitesimally small and thus we can simply remove it from the equation because it's so small to leave us with 2x as the gradient of the x squared graph we just performed a derivative A fundamental idea on calculus, which involves finding the gradient of a function by taking smaller and smaller approximations of a quantity. It's pretty beautiful to see how calculus made these really hard problems so simple. What I showed is an infinitesimally small glimpse of what calculus can actually do.
There's just so much more to explore. Thank you for watching.