if mathematically derivatives are the opposite of integrals, conceptually how is the area under a curve opposite to the slope of a tangent line?
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Responding to your edit. Indefinite integrals are basically defined as anti derivatives, rather than area under a curve. They are defined basically as the function when you take its derivative you end up back where you started. But this isn’t helpful conceptually as it’s just by definition.
But you can also express indefinite integrals as an area under the curve, but you set one of the bounds of the integral to be a variable x’ instead of a constant (the other bound you can just choose an arbitrary constant). Now the slope of this integral is really asking, how fast does the integrated function change with x, or in other words, how fast does the area under the original curve change with x? The additional area you add under the original curve f(x) is just going to be f(x) dx which brings you back to the original function.
Instead of starting at a function f(x) and asking, how is df/dx the opposite of integral f(x) dx, rather think about taking the slope of [ integral f(x) dx] and you see you end up where you started.
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