What is differentiation?

One day while teaching I wanted to find change in a function. The value which it changes by moving a certain part of x axis. I also wanted to find value of any other point on x axis from the value of a known point. Let me show by an example what I wanted to find. For example If y = 2x then

x 1 2 3 4 5 6 10 18
y = 2x 2 4 6 8 10 12 20 36


My first Approach

My first approach was to find the ratio by which the value of the y changes with x. The process which I have already used are "Ratios". In ratios I find the change in one quantity when other quantity changes by 1. The steps to find change in y becomes. For simplicity let us call y = f(x), a function of x.
  1. Find the value of y i.e. f(x) at x1. Which is f(x1).
  2. Find the value of y when x = x1 + 1 i.e. f(x1 + 1).
  3. Find the change in the value of y for unit change in x. i.e. Δy = f(x1+1)-f(x).

Let us use it for y = 2x
  1. y1 = f(x1) = 2x1.
  2. y2 = f(x1 + 1) = 2(x1 + 1).
  3. Find the change in the value of y for unit change in x.
    i.e. Δy = y2 - y1 = f(x1 + 1)- f(x) = 2(x1 + 1) - 2(x1) = 2.
By the use of ratio the value of any x is y/x = Δy/Δx → y = (Δy/Δx)x. Here this formula works very well to find the successive values of y for x.

Now let us look at another function y = x2.

x 1 2 3 4 5 6 10 18
y = x2 1 4 9 16 25 36 100 324

Let us compute the change with the similar method above.
  1. y1 = f(x1) = x12.
  2. y2 = f(x1 + 1) = (x1 + 1)2.
  3. Find the change in the value of y for unit change in x.
    i.e. Δy = y2 - y1 = f(x1 + 1)- f(x) = (x1 + 1)2 - (x1)2 = 2x1 + 1.
Let us compute the change, y = (Δy/Δx)x = (2x + 1)x 
But, here we find that the value of next quantity depends on it and it does not give the correct value. Let us compute some values.
x 1 2 3 4 5 6
y = (2x + 1)x 1 10 21 36 55 78

I thought a little bit and decided to change the value of change to very small. But what should be the change. I decided to take it very small and tending to zero as it becomes a general value. Then the steps to find the change becomes

  1. Find the value of y i.e. f(x) at x. Which is f(x).
  2. Find the value of y when x = x + δx i.e. f(x + δx). Here change in x is very small and equal to δx.
  3. Find the change in the value of y for δx change in x. i.e. δy = f(x + δx)-f(x).
  4. Find the ratio of change in y to change in x. δy/δx = [f(x + δx)-f(x)]/δx.
  5. Find the ratio of change in y to change in x when δx tends to zero.

    lim (δy/δx) = lim [f(x + δx)-f(x)]/δx.
    δx→0          δx→0          
    
  6. Let us call
    lim (δy/δx) = dy/dx 
    δx→0  
    then 
    dy/dx = lim [f(x + δx)-f(x)]/δx.
            δx→0                          
    
When I found this formula I was involved in finding its interpretation. Which I will describe in the next post.



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