WHY IS DERIVATIVE NEEDED IF THERE IS ALREADY UNITARY METHOD FOR CALCULATING RATE? So, here we are taking the slope of the slant line by taking two sample points A and B arbitrarily by the below equation (y2-y1)/(x2-x1) = (3-1)/(3-1) = 2/2 = 1/1 which shows that the slant line has 1 unit rise for every 1 unit run, which holds true for any segment of the line taken arbitrarily (yes, we can see it has hold true for two arbitrary points A and C as shown by the lines explicitly 1 rise for 1 run). Yes, here the situation was forgiving for this unitary method of rate taking because the line of whose slope we were taking was itself uniform. But consider the graph below So, if we proceed with our usual rate calculating method (ie. unitary method) by taking two sample points A and B as first case, then we will find the slope=1/1 ie. 1 unit rise for every 1 unit run . But that’s here we get caught if we just take another sample points A and C because when we move from A to C, there is more than 1 rise for 1 run. So, to avoid this, we chose the two sample points so close to each other that they are almost the same point because that way, the slope we calculate is unique of the point concerned and we solve the issue that slope calculated is representative of every points within the sample points because now our sample points are almost at same place ,or to say in mathematical language, one sample point tends to another in limit or the width of the two sample points (delta x) tends to zero in limit. The width of the interval, which is shown by the two narrow lines, is in fact so small but is in the size shown only for exaggeration for clarity. And now, slope = which is written short-handedly as dy/dx and is called derivative of y wrt x.