I am trying to understand the relationship between the function $f$ and its derivative function $g$, or in other words, $g$ and its antiderivative $f$.
Here is what i understand: the graph of function $g$ plots the tangents of the graph of function $f$ to $x$ input.
If i am not wrong: the graph of function $f$ plots the area under graph of function $g$ to $x$, where $x$ is the lower bound of the area under the graph $g$.
Now, given $g(x) = \frac1x$ and $f(x) = \ln |x|+C$, i plot both graphs using Desmos: https://i.stack.imgur.com/cKeDh.jpg
Graph of function g does indeed map the slopes of tangents of function $f$ at each $x$ value. However, the area under graph of function $g$ is clearly not mapped by graph of function $f$: when $X > 0$, area under graph of function $g$ is always +ve, but graph of function $f$ will always give some -ve output. Why is this so?
I would greatly appreciate your assistance!
If $f = g'$, then the Fundmental Theorem of Calculus says that $$\int_a^b f = g(b) - g(a).$$ Note that subtracting these values will remove the free parameter $C$.
Basically, what I'm saying is that the anti-derivative doesn't just give you the area, but the change in value in the anti-derivative will give you the (signed) area under the curve.
It's also worth noting that the derivative doesn't really plot the tangent line per se, but plots the gradient of the tangent line at the given $x$ value.