What is Differentiation?
Differentiation tells us how fast something is changing. We can say that the rate of change of anything will be considered as a derivative. The rate of change has two types i.e continuous rate of change and instantaneous rate of change.
But first of all, we have to know what is the rate of change and later on we can discuss the instantaneous and average rate of change. These concepts helps you to understand the concept of differentiation/derivative calculus.
Let’s see what is the rate of change?
Rate of change?
Whenever something will change it happens with respect to someone else.
Let’s say you are observing a moving car then the change in the position of the car is with respect to you. The change happening with the relationship of two things, one is who changed the position other is from whom with respect to change occurs.
$$ Relation = \frac{\Delta something\;changes} {\Delta w.r.t\;which\;change\;happens} $$
Example
You are traveling back to your home and you travelled 100km but there is still 200km remaining.
Let’s suppose there is nothing like “speed” in the universe then it’s difficult for you to imagine how much time you will be able to reach home.
But it is too easy to tell if you get a relation (as we discussed in the above paragraph) the change in distance with respect to the change in time, means how fast distance is travelled in relation to time.
Now we can define a relation called speed which is between change in distance with respect to change in time is given as follow:
$$ Speed = \frac{\Delta Change\;in\;distance}{\Delta Change\;in\;time} $$
So, that relation which we elaborate by example is known as the rate of change that one thing changes with respect to another.
Now you can detect that the rate of change is not a single entity but a relation between two entities.
Most of the happenings in our daily life have a relation or rate of change with respect to others which you can easily detect by using the relation formula.
Now we will discuss what is the instantaneous rate of change and why we need instantaneous change.
Instantaneous rate of change:
For an instantaneous rate of change, let’s take again an example of a car moving from point 1 to point 2. For speed, we calculate how much distance from point 1 to point 2 is travelled and change in time from the initial point to the ending point.
The two key points that you have to notice are
- You always need two points when you calculate the change.
- The second thing is that the speed that you calculate from the speed relation is the average speed between these two points.
Now suppose the car collides with some other car in-between point 1 to point 2, then can we determine the speed of a car at accident equals to average speed? obviously not, because the speed of the car is different at that exact time of the accident.
In general, the speed of the car at an exact time or exact instant is called the instantaneous rate of change. More generally, the instant rate of change is called the Derivative.
Derivative calculus:
We discussed how instantaneous change is known as a derivative, but how to find the speed at an exact instant and what will be the distance at the exact time is complex but not impossible for mathematicians to find.
We can explain it in three very simple steps:
In the Step 1, we choose a second point which is very close to our instant point.
We have only point 1 e.g t=5 sec, then we choose a second point that is very close to point 1 e.g t=4.999 sec.
Now In Step 2, we will find the average speed at this very close and short interval of change.
The Step 3 is very important in which we learn to select point 2 as close to our instant point so that the change between these points is nearly zero but not exactly zero. Because, if the change will be zero we can’t divide anything by zero.
In this way, mathematicians discovered such a small real number that are not zero and can be used for dividing. In general, for calculating the speed at an instant, calculate the average speed of such a small interval which almost equals zero.
i.e
$$ Speed(at \;\; time=5 ) = \lim\limits_{ \Delta time \to 0} \frac{ \Delta Distance}{ \Delta time} $$
Power of generalization:
We know instantaneous speed is a change of speed at an instance of time. Now we use some general variables e.g "y" and "x" with a function
$$ y=x^2 $$
In this function, we simply identify that how "y" is changed if we apply the change in "x".
So for derivatives, we simply follow the same 3 steps which we follow in instant speed.
In the Step 1, we suppose a second point like x+Δx which is a very short interval. In the Step 2, we will calculate the average rate of change at that interval(x+Δx). $$ \frac{Δy}{Δx} = \frac{(x+Δx)^2- x^2}{Δx} \; \implies \; \frac{(x)^2+2xΔx}{Δx} $$
Now in the Step 3, We further follow two steps
We can say if "x" is not equal to 0, then we can divide the rate of change by it
$$ \frac {Δy}{Δx} = Δx+2x $$
Also, we consider that "x" is a very small interval, so the Δx~0
Now,
$$ \frac {Δy}{Δx} = 2x $$
And
$$ \frac{dy}{dx}= \lim\limits_{ \Delta x \to 0}\frac {Δy}{Δx} =2x $$
Now in general you can say that, If we calculate the average rate of change for a very short interval which is not exactly zero but x~0, then you can find the instantaneous rate of change which is called the derivative of that function.
In dy/dx variables form, you can call it the derivative of "y" with respect to "x".