sensing the link between the tangent problem and the area problem

While we know that the fundamental theorem of calculus (FTC) establishes the link between differentiation and integration, what could have inspired Newton or  Leibniz to formulate the theorem?

It is said that even before the FTC was derived, mathematicians have already sensed a link between the tangent problem and the area problem, but what was this intuition that hovered behind the FTC?

Consider that the differentiation of displacement would give velocity, then summing the area of the velocity graph would ‘intuitively’ give the amount of distance traveled, ie. displacement. One can also consider that the accumulation of wealth over time would simply be the sum of the rate of accumulation at each point of time. (or the total amount of haze over time = the sum of PSI concentration at each point in time).

For some idea of the FTC: Let A(x) be the area obtained under the function f(t) starting from x=a.

A(x+h)-A(x) \approx f(t)h

\frac{A(x+h)-A(x)}{h} \approx f(t)

A'(x) = f(t) \blacksquare

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