Everything is changing with respect to time.If we see ourselves also, our body is changing as we are growing up.If we want to compare two or more things that which one is changing faster and whichone slower,then we have to define a specific quantity which deals with this problem. For example speed,it is the rate of change of distance with respect to time.We can compare that which car is moving faster by just knowing the speed of both cars.
So, we define speed as distance covered in a specific time or: speed=distance/time.Similarly,we define other quantities.
Now,we know that in a city like Delhi, there is a problem of traffic jam!So, a car’s speed keeps changing,for example,for first 15 min,a car’s speed is 20 km/hour and for the next 5 min,it changes to 25 km/hour.
What about if we want the exact speed of the car at 21 th second?
Forget this problem and let’s take a new function, say: f(x)=2x.
Now, f(1)=2& f(2)=4
what’s the difference between these two functions??
yes,it’s f(2)-f(1) = 4-2=2
what’s the difference between the two no.s?
what if we divide these quantities??
yes, our answer is 2÷1=2 .So, f(2)-f(1)/(2-1) = 2.
We have calculated the rate of change of function with respect to x!!
we can also write it as: f(x2)-f(x1) / (x2-x1).
Similarly for calculating speed between any two times,we just follow the above method as speed isa function of time. So we can find speed between any two moments,
But what about at a specific instant as our main question was??
If the difference between these two times becomes zero,we can easily find our solution but the problem is that our formula will take the form of 0/0 which is not defined!!
So, what can we do now??
For these problems specifically, limit was introduced!
We can’t make the difference exactly zero but we can set a limit to our function where the difference tends to zero.
Now ,it’s very easy to find the exact speed at nth second.
So our formula will now become:
THIS is nothing but the derivative of the function!!!
So, now we can define derivative as:
The derivative of y with respect to x is defined as the change in y over the change in x, as the distance between x1 and x2 becomes infinitely small .