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.
- Find the value of y i.e. f(x) at x_{1}. Which is f(x_{1}).
- Find the value of y when x = x_{1} + 1 i.e. f(x_{1} + 1).
- Find the change in the value of y for unit change in x. i.e. Δy = f(x_{1}+1)-f(x).
Let us use it for y = 2x
- y_{1} = f(x_{1}) = 2x_{1}.
- y_{2} = f(x_{1} + 1) = 2(x_{1} + 1).
- Find the change in the value of y for unit change in x.
i.e. Δy = y_{2} - y_{1} = f(x_{1} + 1)- f(x) = 2(x_{1} + 1) - 2(x_{1}) = 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 = x
^{2}.
x |
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2 |
3 |
4 |
5 |
6 |
10 |
18 |
y = x^{2} |
1 |
4 |
9 |
16 |
25 |
36 |
100 |
324 |
Let us compute the change with the similar method above.
- y_{1} = f(x_{1}) = x_{1}^{2}.
- y_{2} = f(x_{1} + 1) = (x_{1} + 1)^{2}.
- Find the change in the value of y for unit change in x.
i.e. Δy = y_{2} - y_{1} = f(x_{1} + 1)- f(x) = (x_{1} + 1)^{2} - (x_{1})^{2} = 2x_{1} + 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
- Find the value of y i.e. f(x) at x. Which is f(x).
- 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.
- Find the change in the value of y for δx change in x. i.e. δy = f(x + δx)-f(x).
- Find the ratio of change in y to change in x. δy/δx = [f(x + δx)-f(x)]/δx.
- 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
- 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.