Recipe

Differentiate an accumulation function

Use when: you're asked for H'(x) where H(x)=\int_a^{g(x)} f(t)\,dt and the upper limit is a function of x, not x itself

Goal: a formula for H'(x), found without ever computing H(x) in closed form

Taught in FTC Part 1: the chain rule twist

1. 1

   Name the integrand f(t) and the upper-limit function g(x) separately.

   Example

   For H(x)=\int_0^{x^2}\sin(t)\,dt: the integrand is f(t)=\sin t, and the upper limit is g(x)=x^2.

   Why: the twist ahead treats these two pieces differently, so keeping them separate from the start avoids conflating "what's being integrated" with "how far the integration goes."

2. 2

   Substitute g(x) for t in f(t).

   Example

   f(g(x))=\sin(x^2).

   Why: FTC Part 1 says the accumulation function's rate equals the integrand evaluated right at the current endpoint, and here the endpoint is g(x), not x itself.

3. 3

   Multiply the result of Step 2 by g'(x).

   Example

   g'(x)=2x, so H'(x)=\sin(x^2)\cdot2x=2x\sin(x^2).

   Why: this is the chain rule doing its usual job — H depends on x only through g(x), so its rate of change picks up a factor of how fast g(x) itself is changing.

4. 4

   Plug the requested x-value into the formula for H'(x) from Step 3.

   Example

   At x=1: H'(1)=2(1)\sin(1)=2\sin(1)\approx1.68.

   Why: Steps 1–3 already built the general formula. Evaluating it at one point is now just arithmetic.