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The notation $df/dx$ motivates thinking of tiny vertical changes paired with tiny horizontal changes, and informally canceling $dx$ helps intuition. However, it is misleading because the derivative is defined by a limit, not by dividing two ordinary numbers.

Conditions: Interpreting the differential notation in the context of the theorem.; Understanding the formal definition of the derivative.

The notation $df/dx$ motivates thinking of tiny vertical changes paired with tiny horizontal changes, and informally canceling $dx$ helps intuition. However, it is misleading because the derivative is defined by a limit, not by dividing two ordinary numbers.

Conditions: Interpreting the differential notation in the context of the theorem.; Understanding the formal definition of the derivative.

Continuity ensures that as the interval width $h$ approaches zero, the average value of $f$ over $[x, x+h]$ converges to the instantaneous value $f(x)$. Without continuity, the local behavior might oscillate wildly or have jumps, preventing the limit of the difference quotient from settling on a single well-defined value $f(x)$.

Conditions: $A(x) = \int_a^x f(t) dt$; $f$ is continuous at $x$

Continuity ensures that as the interval width $h$ approaches zero, the average value of $f$ over $[x, x+h]$ converges to the instantaneous value $f(x)$. Without continuity, the local behavior might oscillate wildly or have jumps, preventing the limit of the difference quotient from settling on a single well-defined value $f(x)$.

Conditions: $A(x) = \int_a^x f(t) dt$; $f$ is continuous at $x$

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 $x$.

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 factor is essential because the Fundamental Theorem of Calculus combined with the Chain Rule requires multiplying the integrand evaluated at the upper limit by the derivative of that limit. Without this factor, the rate of change of the accumulated area with respect to the outer variable $t$ would be incorrect.

Conditions: The upper limit is a function of $t$, not just $t$ itself.; The integrand is continuous.

The factor is essential because the Fundamental Theorem of Calculus combined with the Chain Rule requires multiplying the integrand evaluated at the upper limit by the derivative of that limit. Without this factor, the rate of change of the accumulated area with respect to the outer variable $t$ would be incorrect.

Conditions: The upper limit is a function of $t$, not just $t$ itself.; The integrand is continuous.