Calculus I · Unit 3A · lesson

Variable Limits and the Chain Rule

Concept

Learning objectives

Differentiate integrals whose bounds are functions of the differentiation variable.

Variable Limits and the Chain Rule

Explanation

An accumulation endpoint can itself be changing

The Fundamental Theorem handles $\int_a^x f(t)\,dt$ , but many problems use an endpoint such as $g(x)$ . In that case the accumulated total changes for two reasons: the endpoint moves, and the speed of that movement is $g'(x)$ . The derivative is therefore $f(g(x))g'(x)$ , which is the ordinary chain rule applied to an accumulation function.

Lower limits contribute a minus sign because moving the lower endpoint to the right removes accumulation rather than adding it. With both endpoints variable, differentiate the upper contribution and subtract the lower contribution. A reliable approach is to name the accumulation function first, apply the Fundamental Theorem, and then apply the chain rule visibly rather than attempting to remember a pile of signs from memory.

G(x)=\int_a^{g(x)}f(t)\,dt,

then

G'(x)=f(g(x))g'(x).

The FTC supplies the outer derivative; the chain rule supplies the factor $g'(x)$ .

If both limits vary,

\frac{d}{dx}\int_{u(x)}^{v(x)}f(t)\,dt =f(v(x))v'(x)-f(u(x))u'(x).

Worked example

Two moving boundaries

For

H(x)=\int_{x^2}^{\sin x}e^{t^2}\,dt,

H'(x)=e^{\sin^2x}\cos x-e^{x^4}(2x).

No elementary antiderivative of $e^{t^2}$ is required.

Interactive checku3a-variable-limits-01

Find $\frac{d}{dx}\int_0^{x^2}\sin t\,dt$ .

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Show hint

Evaluate the integrand at $x^2$ , then multiply by $2x$ .

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After the explanation

Use the section idea

Reading lens

Use the Fundamental Theorem as the bridge between accumulation functions, local rates, and endpoint evaluation.

Mental model

Differentiating a running total recovers its current integrand, while evaluating an accumulated total subtracts antiderivative endpoint values.

Decision

Separate FTC Part I, FTC Part II, net change, and variable-bound chain-rule tasks before manipulating notation.

Common trap

Forgetting a chain-rule factor at a variable bound, reversing endpoint subtraction, or adding +C to a definite value.

Check yourself

Can you state which part of the theorem applies and why its hypotheses and bounds fit?

Source & rights

Original instruction with traceable references.

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