As said our goal is to bring both dx and dy tend to zero not just dx or dy.

Our goal is to take a limit, which is a very precisely defined thing, nothing more and nothing less.

CORRECT – Take two points A and C on a curve of x^2. As mentioned multiple times there is a point beyond which dx can’t be tended to zero as dy diminish or reaches zero earlier than dx because the length of dy < length of dx when Tan theta < 45 degrees. Although there is dx but no dy means no slope. At this point, limiting dx to zero would be meaningless as we lost dy and slope.

Another try but in detail.

The gradient of the graph of y = x2 at any point is twice the value of x thereat. The process of finding the derivation of a gradient / slope of a function y=f(x) or y = x2 is as follow.

Pick any two points A and B close to each other on the curve of y =x2. The coordinates of A on the curve are (x, y) or (x, x2). Add Δx at A as usual. When x increases by Δx, then y increases by Δy. The x changes from x to (x +Δx) while y changes from y to (y + Δy) or f(x) to (x+Δx)2. Thus the x and y coordinates of B on the curve are (x + Δx, y + Δy) or ([x+Δx, (x+Δx)2]. Now the instantaneous rate of change is given by

Δy/ Δx = [(x + Δx)2 – x2] / [x + Δx - x]

Δy/ Δx = [x2 + Δx2+2xΔx − x2] / Δx

Δy/ Δx = [2x + Δx] / 1

Reduce Δx close to zero by taking limit (Δx to dx and Δy to dy)

dy /dx = 2x + dx

dy /dx = 2x------Eq1 OR

dy = 2x.dx --Eq2

ABC is an infinitesimal triangle made by dx, dy, and hypotenuse or slope of tangent where point A and C are always on the curve. Length of AB = Base = dx, Length of BC = Perpendicular= dy and Length of Hypotenuse = AC. Angle CAB or BAC is the slope of a tangent

According to the aforementioned Eq1 or Eq2-

• dy/dx is directly proportional to x or angle CAB is directly proportional to x.

• dx is indirectly proportional to x OR x is inversely proportional to dx

• dy is directly proportional to x.dx or dx

The length of dx > dy when Angle CAB < 45 degrees The length of dx = dy when Angle CAB = 45 degrees The length of dx < dy when Angle CAB > 45 degrees

The proportionality of both the angle CAB and dy with x are in contradiction with the proportionality of x and dx in the triangle ABC after probing the equation of dy/dx = 2x beyond its derivation on a graph of y = x2. When x increases; dx decreases, dy increases, and angle CAB increases. This means AC also increases and ultimately SECANT when x increases. Our goal is to bring dx, dy and AC to zero (not away from zero either positively or negatively - Point C has to be on the curve or secant to tangent by reducing them close to zero but here dx heads toward zero but dy and AC increases when x increases on axis mathematically.

Although the difference in the length of dx and dy can be noticed clearly on the graph if we examine the triangle ABC at two different points for a gradient (dy/dx), say at when an angle BAC = 0.1 degrees and 89.9 degrees on the curve.

RISE = dy = 2x and RUN = dx = 1 (always constant) in a GRADIENT of 1 in 2x which we obtained from the Eq1 of dy/dx=2x /1 at any point on the curve when there is no difference between secant and tangent – No idea how do we get dy/dx = 2x.dx but above said contradiction may be due to the introduction of another curve of y =(x+dx)2 at a point where we seize x or y=x2 deliberately and introduce delta x OR when function y = f(x) changed to y=f(x+Δx)2. The value of x has reached to its maximum value instead of unlimited when a curve y=x2 doesn’t continue anymore at a point where we introduce delta x or dx as y=x2 and y =(x+dx)2 are two different types of curve (two different functions).

Further, integration is the reverse process of differentiation. Although delta x or dx is ignored during the process of derivation of dy/dx because of their small values but we can’t ignore them in the process of integration which makes a lot of difference in summation. They can’t be disappeared forever and should resurface during the process of integration or summation.

Similarly, dy is the small vertical change in y, therefore, we take the sum of all the small vertical lengths [dy(s)] not the whole slice or y-coordinate(s) from zero to its value on the curve when we integrate both sides of the equation of dy = 2xdx but it turns into a function of x2 or area under the graph – no idea how but the summation of vertical lengths on a graph gives vertical length only not curve?