Differentiation measures change. Integration rebuilds it — but how? Connect the two through a single example: if , the sign of the derivative tells us which way the original function moves, its magnitude tells us how steeply, and integrating that same rate reconstructs the function itself, up to an unknown vertical shift.

Concept The rules for reading a derivative

— the original function is decreasing

— the original function is increasing

— a horizontal tangent at that point

throughout an interval — the original function is constant there

Example Reconstructing a parabola from f′(x) = x

On the right of the y-axis, is positive, so the original function is increasing — and since grows as we move right, it becomes increasingly steep. On the left, is negative: the function decreases, steeply at first, then flattens as the derivative approaches zero near the origin. At the derivative is zero, giving a horizontal tangent — the vertex of a parabola.

Far left — large negative derivative, falling steeply.

Near zero — derivative shrinking toward zero, flattening out.

Vertex — derivative exactly zero, momentarily flat.

Right — derivative growing positive, rising ever more steeply.

Every feature of the parabola — where it falls, where it flattens, where it rises — was predicted just by reading .

Concept Why every integral needs + C

The derivative of , of , and of are all exactly — sliding a graph up or down never changes its slope. The constant represents every possible vertical position of the original function.

The derivative alone can never fix that position: one known point on the actual graph is what pins down the exact value of .

Concept The derivative of accumulated area

Now accumulate the signed area under , starting from 0. On the right, each new vertical slice is taller than the last, so the accumulated area grows faster and faster — that accumulated function is again . On the left, is negative, but integrating from 0 toward a negative value also reverses the direction of the limits: two negatives cancel into a positive accumulated value, tracing the very same parabola on that side too.

This is the heart of the connection: the rate at which accumulated area changes equals the current height of the function being integrated.

Note A constant derivative, for contrast

If the derivative is always 4, every equal step along the x-axis adds the same rectangular slice of area, so the original function is a straight line: . A derivative of zero everywhere adds no area at all — the original function is constant.

Summary

The sign of a derivative tells the direction of change; its magnitude tells how steeply.
Reading alone predicts the full shape of the parabola .
Every integral carries a , since vertical shifts never change slope — one known point pins down its value.
The derivative of an accumulated-area function equals the height of the function being integrated, at every point.
A constant derivative accumulates into a straight line; a zero derivative accumulates into nothing at all.

The Derivative of Accumulated Area