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Lesson 07 · Core idea two · 8 min

Antiderivatives

An antiderivative runs the derivative backwards: given the rate of change, recover the original quantity. If you know the speed at every moment, can you rebuild the journey? Yes — that's antidifferentiation.

Build the intuition

Differentiation, rewound

F is an antiderivative of f when F′ = f. Since (x²)′ = 2x, an antiderivative of 2x is x². Every derivative fact you know is already an antiderivative fact read right-to-left.

F′(x)=f(x)F'(x) = f(x)

The +C: one rate, many originals

x², x² + 5, and x² − 100 all have derivative 2x — vertical shifts don't change slopes. So from the rate alone, the original is only known up to a constant. We write +C to hold the door open for the whole family.

∫2x dx=x2+C\int 2x\,dx = x^2 + C

Why the missing constant is physical

Knowing your speed all day tells you how far you traveled — but not where you started. The starting point is C. Given one extra fact (“at t = 0 I was home”), C snaps into place and the original is fully recovered.

See it move

InteractiveWhy +C exists
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Every curve x² + C has slope 2 at x = 1 — the derivative can't tell them apart, so the antiderivative must carry a +C.

Slide C: the curve glides up and down through a family of parallel originals, every one wearing the same slopes.

A worked example

Recover position from velocity

  1. A drone climbs at v(t) = 3t² meters per second.

  2. Antidifferentiate (reverse power rule):

    h(t)=t3+Ch(t) = t^3 + C

  3. It launched from a 2 m platform, so h(0) = 2 gives C = 2:

    h(t)=t3+2h(t) = t^3 + 2

  4. Rate plus starting point — the full story, rebuilt.

Out in the world

Fitness trackers

Your watch measures acceleration hundreds of times per second. From that it rebuilds speed, then distance — antidifferentiation in real time. The arithmetic in your watch is the math in this lesson.

Common confusion, cleared

“The +C is decoration teachers insist on.”

It's real information honestly missing from the rate. Drop it and you've silently claimed the journey started at zero.

“A function has one antiderivative.”

It has infinitely many — a whole family of vertical shifts. They differ only by where they start.

Recap

  • Antiderivative: F with F′ = f — the derivative, undone.
  • Always +C: shifts share slopes, so the rate can't see the start.
  • One known value (an initial condition) pins down C.