AP® Calculus AB · Unit 6 · Topic 6.4
Differentiating an Accumulation Function
If g(x) is the integral of f(t) dt from a to x, then g'(x) = f(x). Differentiating an accumulation function just hands back the original function, evaluated at x.
An accumulation function g(x) measures the running total of f(t) from some starting point a up to x. As x grows a little, g picks up one more sliver of area, and that sliver's size is controlled by f(x) itself, the height of the function right at the edge you just moved past. So the rate at which g accumulates area is exactly f(x). That's the second Fundamental Theorem of Calculus: g'(x) = f(x). Differentiation and integration undo each other, and this is the cleanest possible version of that fact — no antiderivative formula needed, no algebra, just read off the integrand.
The lower limit a never shows up in the answer. Changing a from, say, 0 to 5 shifts g(x) by a constant amount, the fixed chunk of area between 0 and 5. Shifting a function by a constant doesn't change its slope, so g'(x) comes out identical no matter what a is. All that matters is the integrand and the variable upper limit.
When the upper limit is something other than plain x, like x3 or x2, chain rule kicks in: differentiate the accumulation function with respect to the upper limit, then multiply by the derivative of that upper limit. This topic sits right after you've built integrals as limits of Riemann sums, and it's the bridge to the first Fundamental Theorem, where you actually evaluate definite integrals by finding antiderivatives rather than differentiating them back.
Key terms
- Accumulation function
- A function g(x) defined as a definite integral of f(t) from a fixed lower limit to a variable upper limit x.
- Second Fundamental Theorem of Calculus
- The rule stating that if g(x) is the integral of f(t) dt from a to x, then g'(x) = f(x).
- Integrand
- The function being integrated, written inside the integral as f(t).
- Chain rule
- A differentiation rule used when the upper limit of the accumulation function is itself a function of x, not just x.
Watch it explained
- How it worksHow does the second Fundamental Theorem undo an integral in one step?
- Worked exampleFind the derivative of the integral of sin(t2) dt from 1 to x.
- Worked exampleFind the derivative of the integral of et dt from 0 to x3 using the chain rule.
- Common mistakeWhy isn't the derivative of an accumulation function changed by the lower limit?
Check yourself
End of topic.