As we know that tangent is the rate of change of a function and by the definition of a tangent on a curve, we know that it touches the curve at a single point but by the formula of differentiation, we find two points on the curve and try to minimize the difference between them but still we have the slope touching at least two points.
In a graphical representation:
- Text in the image:
- h->0
- The actual slope (of the tangent)
- The formulated slope (i.e. according to the definition of the differential)
The short answer is no. The definition of tangent to a curve at a point on the curve in Wikipedia (for the case where the curve is the graph of a function $ f $) says it is a line through that point with the same slope as the curve at that point, and the slope is defined as the differential. I.e. the slope of the tangent is by definition the result of differentiation.
The first illustration in that Wikipedia article also shows a tangent passing through another point on the curve, refuting your claim that the tangent only passes through one point on the curve.
A longer and perhaps more useful answer would help you to understand the misconceptions implicit in your question, but that is harder to supply, as your reasoning is not altogether clear. To be extended: I shall try to assemble a fuller answer when I have a little more time