Accumulation derivative
Differentiate an accumulation function with a variable upper bound.
Accumulation derivative is one of 7 integration formulas in the calculus section of this library, and it is used at ap · university level.
Why accumulation derivative works
The accumulation function records how much area has been collected so far. Nudging the upper limit a little to the right adds one thin sliver whose width is that nudge and whose height is the function value there, so the collected total grows at exactly that height.
What each symbol means
$t$ is a dummy integration variable and $x$ is the upper limit.
Accumulation derivative: when it holds
$f$ is continuous near $x$; apply the chain rule when the bound is $g(x)$.
When it stops applying
It is stated for an upper limit of x alone. When the upper limit is some other expression you must also multiply by that expression's derivative, so with an upper limit of x squared the accumulation of cos t differentiates to 2x times cos of x squared.
Accumulation derivative: a worked example
$\frac{d}{dx}\int_1^{x^2}\cos t\,dt=2x\cos(x^2)$.
The mistake to avoid
What people do: Finding an antiderivative, plugging in the upper limit, and then differentiating a second time.
Why it goes wrong: The theorem already delivers the derivative in a single step, so the extra work returns the derivative of the integrand rather than the integrand itself.
Do this instead: Simply swap the dummy variable for the upper limit and stop. Nothing else is required.
Accumulation derivative: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $t$ is a dummy integration variable and $x$ is the upper limit.
- Check the conditions before substituting. $f$ is continuous near $x$; apply the chain rule when the bound is $g(x)$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Integration
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where accumulation derivative comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Fundamental Theorem of Calculus — the lesson behind this formula: connect accumulated area with instantaneous change.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about accumulation derivative
Why does the lower limit never appear in the answer?
Changing it shifts the whole accumulation by a fixed amount, and a fixed shift has zero derivative, so it cannot influence the rate of growth.
What if the variable sits in the lower limit instead?
Swapping the two limits reverses the sign of the integral, so the derivative comes out as the negative of the function value at that limit.
Why is a different letter used inside the integral?
One letter runs across the interval being added up while the other marks the right-hand end of that interval. Using the same symbol for both would confuse a moving point with a fixed one.
Does the integrand have to be positive?
No. Where it is negative the accumulation decreases instead of growing, and the theorem reports that correctly as a negative rate of change.