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The derivative $A'(x)$ equals $f(x)$ because moving the endpoint from $x$ to $x+h$ adds a thin strip whose area increment is approximately $f(x)h$. Dividing this increment by $h$ and taking the limit as $h$ approaches zero yields the local average height, which continuity ensures approaches the exact endpoint height $f(x)$.

Conditions: The function $f$ is continuous at $x$.; $A(x)$ is defined as the accumulation of $f$ from a fixed lower limit to $x$.

The derivative $A'(x)$ equals $f(x)$ because moving the endpoint from $x$ to $x+h$ adds a thin strip whose area increment is approximately $f(x)h$. Dividing this increment by $h$ and taking the limit as $h$ approaches zero yields the local average height, which continuity ensures approaches the exact endpoint height $f(x)$.

Conditions: The function $f$ is continuous at $x$.; $A(x)$ is defined as the accumulation of $f$ from a fixed lower limit to $x$.