Mathematics

The Fundamental Theorem of Calculus tells us that when you define a function as an integral with a variable upper limit, its derivative simply gives you back the integrand evaluated at that point. Here, $g(x)$ accumulates area under $f$ from a fixed left endpoint to $x$, so $g'(x)$ measures how fast that accumulated area grows—which is exactly $f(x)$ itself.

The Fundamental Theorem of Calculus tells us that when you define a function as an integral with a variable upper limit, its derivative simply gives you back the integrand evaluated at that point. Here, $g(x)$ accumulates area under $f$ from a fixed left endpoint to $x$, so $g'(x)$ measures how fast that accumulated area grows—which is exactly $f(x)$ itself.