What Is derivative? (Part - 1)

Derivative is a simple thing. It’s just the difference of a variable. But how?

Think of time, let a person started a journey towards bus stand at time t1 = 3 and reached at t2 = 7. In this case the derivative dt = t2-t1 = 7-3 = 4. Easy!

But in real life things are not that easy. Because we have do derive the difference of a variable with respect to another variable unlike our time example where there was only one variable. For example we can see the equation y = x. Here we have to find the difference in y with respect to the change in variable x. In other word we have to find the derivative of y with respect to x which is dy/dx. Fortunately for us this is very easy.

Pretty easy right. But we are going to prove it so that we can use the same theory to prove more complex derivatives. So we are going to plot this function into graph. Let’s find some value of x and values of y for the respective value of x below.

x	y = x
0	0
1	1
2	2
3	3
4	4
5	5
6	6
7	7
8	8
9	9
10	10

Now let’s draw the graph. It should be easy

For ever value of x the value of y is the same. Remember what we have to find. We will find the difference in y with respect to the change in x. so let’s pin two points in x axis and also pin corresponding two points in y axis.

So the change in x is from 2 to 6 which is 6-2 = 4 and the corresponding difference in y is from 2 to 6 which is 6-2 = 4. Here the derivative of y means with respect to x (dy / dx) will be slope of the triangle that has been formed. Let’s see that triangle below.

Here the height of the triangle is the difference in y = 6-2 = 4 and the base of the triangle is the change in x = 6-2 = 4. Look at below.

We know the slope of a triangle is equal to the height of the triangle divided by its base. So slope = height / base = 4/4 = 1. So dy/dx = difference in y / change in x = 4/4 = 1. Easy right.

Let’s change the equation. We will now use the equation y = 2x. Here derivative of y with respect to 2x will be dy/dx = 2. Like before we will plot this function into graph. Let’s find some value of x and values of y for the respective value of 2x below.

x	y = 2x
0	0
1	2
2	4
3	6
4	8
5	10
6	12
7	14
8	16
9	18
10	20

Now let’s draw the graph. It should be easy

Like before we will pin two points in x axis and also pin corresponding two points in y axis.

So the change in x is from 2 to 6 which is 6-2 = 4 and the corresponding difference in y is from 4 to 12 which is 12-4 = 8. Here the derivative of y means with respect to x (dy / dx) will be slope of the triangle that has been formed. Let’s see that triangle below.

Here the height of the triangle is the difference in y = 12-4 = 8 and the base of the triangle is the change in x = 6-2 = 4. Look at below.

Again, we know the slope of a triangle is equal to the height of the triangle divided by its base. So slope = height / base = 8/4 = 2. So dy/dx = difference in y / change in x = 8/4 = 2. Easy right.

Remember this is only true for straight lines not for curves like the equation y = x². Here derivative of y with respect to x² will be dy/dx = 2x. But here we cannot find the slope of the curved line. To see that like before we will plot this function into graph. Let’s find some value of x and values of y for the respective value of x² below.

x	y = x2
0	0
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100

Now let’s draw the graph. It should be easy

Like before we will pin two points in x axis and also pin corresponding two points in y axis.

So the change in x is from 4 to 8 which is 8-4 = 4 and the corresponding difference in y is from 16 to 64 which is 64-16 = 48. Here the derivative of y means with respect to x (dy / dx) will be slope of the triangle that has been formed. Let’s see that triangle below.

Here the height of the triangle is the difference in y = 64-16 = 48 and the base of the triangle is the change in x = 8-4 = 4. Look at below.

We know the slope of a triangle is equal to the height of the triangle divided by its base. So slope = height / base = 48/4 = 12. We can see the slope is also curved. So this formula will only work for straight line. So you can use derivative to find slopes of curved lines but cannot use this graph formula to find the slope.