---
title: "Term-by-term Differentiation — AP Calc BC Definition"
description: "Term-by-term differentiation means taking the derivative of a power series one term at a time, keeping the same radius of convergence. A core BC Unit 10 skill."
canonical: "https://fiveable.me/ap-calc/key-terms/term-by-term-differentiation"
type: "key-term"
subject: "AP Calculus AB/BC"
unit: "Unit 10-infinite-sequences-and-series-bc-only"
---

# Term-by-term Differentiation — AP Calc BC Definition

## Definition

Term-by-term differentiation is the process of differentiating a power series by applying the power rule to each term individually; the resulting series represents the derivative of the original function and keeps the same radius of convergence (though endpoint behavior can change).

## What It Is

Term-by-term differentiation is exactly what it sounds like. If a [function](/ap-calc/unit-1/defining-continuity-at-point/study-guide/JbsR9iQfAzCznNOCG6JK "fv-autolink") is written as a [power series](/ap-calc/unit-10-infinite-sequences-and-series-bc-only/radius-interval-convergence-power-series/study-guide/EPL28ohsnC4hDfyEPMhy "fv-autolink"), you can find its derivative by differentiating each term separately, just like differentiating a polynomial. A power series is basically an infinite polynomial, and this rule says you're allowed to treat it that way.

For example, if f(x) = 1 + x + x² + x³ + ... (the geometric series for 1/(1−x)), then f′(x) = 1 + 2x + 3x² + ... and that new series equals 1/(1−x)². The big theoretical payoff is that the differentiated series has the **same [radius of convergence](/ap-calc/key-terms/radius-of-convergence "fv-autolink")** as the original. The interval of convergence can shrink at the endpoints, though, so you may need to re-check those. This is a BC-only idea that lives in Unit 10 with power series and Taylor series.

## Why It Matters

This concept belongs to AP Calculus BC Unit 10 ([Infinite Sequences and Series](/ap-calc/unit-10-infinite-sequences-and-series-bc-only "fv-autolink")), specifically the part of the CED about representing functions as power series. The exam expects you to build new series from known ones instead of computing every Taylor coefficient from scratch. Term-by-term differentiation is one of the two main tools for that ([term-by-term integration](/ap-calc/key-terms/term-by-term-integration "fv-autolink") is the other). It's how you show, for instance, that the derivative of the sin x series is the cos x series, and it shows up constantly in series-manipulation FRQs where you're handed a Maclaurin series and asked for the series of its derivative.

## Connections

### [Term-by-term Integration (Unit 10)](/ap-calc/key-terms/term-by-term-integration)

The mirror-image operation. Where differentiation turns the [series](/ap-calc/unit-10-infinite-sequences-and-series-bc-only/defining-convergent-divergent-infinite-series/study-guide/CIVFHStGQM90EJ4GtIDB "fv-autolink") for 1/(1−x) into the series for 1/(1−x)², integration turns it into the series for −ln(1−x). Both keep the radius of convergence, and BC FRQs love asking for one right after the other.

### Power Rule (Unit 2)

Term-by-term differentiation is just the [power rule](/ap-calc/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk "fv-autolink") applied infinitely many times. Each term cₙxⁿ becomes n·cₙxⁿ⁻¹, which is why a power series differentiates as easily as a polynomial does.

### Chain Rule (Unit 3)

When the series is built on something like x² instead of x (say, the series for e^(x²)), differentiating each term quietly uses [the chain rule](/ap-calc/unit-3/chain-rule/study-guide/27HxeRGCYJBjuPWBm1uw "fv-autolink"). Exam questions exploit this with substituted series.

### Product Rule (Unit 2)

A useful contrast. You can't differentiate a product factor-by-factor, but you CAN differentiate a sum term-by-term, even an infinite one. Derivatives split cleanly over addition, and power series are giant sums.

## On the AP Exam

This is BC-only content, so AB students can skip it. On the BC exam it shows up in the Unit 10 series FRQ, which appears nearly every year. A typical setup gives you the Maclaurin series for f(x) (or its first few terms plus the general term) and asks you to write the first four terms and the general term of f′(x). The move is mechanical: apply the power rule to each term, then state the general term carefully, watching how the index shifts. Multiple-choice questions test the conceptual side, especially the fact that the radius of convergence doesn't change when you differentiate, even though convergence at an endpoint can be lost. No released FRQ uses the phrase 'term-by-term differentiation' verbatim, but the skill itself is a series-FRQ staple.

## Term-by-term Differentiation vs Term-by-term Integration

They're inverse operations on the same object. Differentiation multiplies each coefficient by the exponent and lowers the power (cₙxⁿ → n·cₙxⁿ⁻¹), while integration divides by the new exponent and raises the power (cₙxⁿ → cₙxⁿ⁺¹/(n+1), plus a constant C). Both preserve the radius of convergence. The classic mix-up is forgetting the +C when integrating or botching the index shift when differentiating. Check direction first: is the power of x going down (differentiating) or up (integrating)?

## Key Takeaways

- To differentiate a power series, apply the power rule to each term individually, just like differentiating a polynomial.
- The differentiated series has the same radius of convergence as the original series, but convergence at the endpoints may be lost.
- This is BC-only material from Unit 10, used to generate new power series from known ones (for example, differentiating the sin x series gives the cos x series).
- When a series FRQ asks for the series of f′(x), differentiate the given terms AND the general term, watching the index shift carefully.
- Term-by-term differentiation and term-by-term integration are inverse tools; differentiation lowers powers and integration raises them and adds a constant C.

## FAQs

### What is term-by-term differentiation in AP Calc BC?

It's differentiating a power series by taking the derivative of each term separately with the power rule. The result is a new power series that represents the derivative of the original function and has the same radius of convergence.

### Does differentiating a power series change its radius of convergence?

No. The radius of convergence stays exactly the same. What can change is endpoint behavior: a series that converged at an endpoint of its interval might diverge there after differentiation, so re-test endpoints if the interval matters.

### How is term-by-term differentiation different from term-by-term integration?

Differentiation sends cₙxⁿ to n·cₙxⁿ⁻¹ (powers go down), while integration sends it to cₙxⁿ⁺¹/(n+1) plus a constant C (powers go up). Both preserve the radius of convergence, but only integration requires the +C.

### Is term-by-term differentiation on the AP Calculus AB exam?

No. Power series are part of Unit 10, which is BC-only. If you're taking AB, you won't see this; if you're taking BC, expect it in the series free-response question.

### Can you really differentiate an infinite series one term at a time?

Yes, for power series inside their radius of convergence. That's the theorem that makes this legal. A power series behaves like an infinite polynomial within its radius, so the derivative of the sum equals the sum of the derivatives there.

## Related Study Guides

- [Unit 10 Overview: Infinite Series and Sequences](/ap-calc/unit-10-infinite-sequences-and-series-bc-only/review/study-guide/8ol6j4eNEB6GkkametRt)

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