---
title: "Nth Derivative — AP Calc BC Definition & Taylor Polynomials"
description: "The nth derivative f^(n)(x) is a function differentiated n times. On AP Calc BC, it builds every Taylor polynomial coefficient in Unit 10, Topic 10.11."
canonical: "https://fiveable.me/ap-calc/key-terms/nth-derivative"
type: "key-term"
subject: "AP Calculus AB/BC"
unit: "Unit 10-infinite-sequences-and-series-bc-only"
---

# Nth Derivative — AP Calc BC Definition & Taylor Polynomials

## Definition

The nth derivative, written f^(n)(x), is what you get after differentiating a function n times in a row. On AP Calc BC, its value at the center, f^(n)(a), divided by n! gives the coefficient of the nth-degree term in a Taylor polynomial (Topic 10.11).

## What It Is

The nth derivative is exactly what it sounds like. Take the [derivative](/ap-calc/unit-10-infinite-sequences-and-series-bc-only/finding-taylor-polynomial-approximations-functions/study-guide/LszguYzKz0M6GdqTRSr6 "fv-autolink") of f, then take the derivative of that, and keep going n times. The result is written f^(n)(x), with parentheses around the n to show it's a count of differentiations, not an exponent. So f^(1)(x) is f'(x), f^(2)(x) is f''(x), and f^(0)(x) is just the original [function](/ap-calc/unit-1/defining-continuity-at-point/study-guide/JbsR9iQfAzCznNOCG6JK "fv-autolink") f(x).

On the BC exam, the nth derivative is the engine behind Taylor polynomials. The CED's essential knowledge for Topic 10.11 says the coefficient of the nth-degree term in a Taylor polynomial centered at x = a is f^(n)(a)/n!. In other words, each derivative you take captures one more layer of information about how f bends and twists near a, and dividing by n! scales it correctly so the polynomial matches f's value, [slope](/ap-calc/key-terms/slope "fv-autolink"), concavity, and so on at the center. The more derivatives you match, the better the polynomial hugs the function near x = a.

## Why It Matters

This term lives in Unit 10 ([Infinite Sequences and Series](/ap-calc/unit-10-infinite-sequences-and-series-bc-only "fv-autolink"), BC only), specifically Topic 10.11. It directly supports learning objective [AP Calc](/ap-calc "fv-autolink") 10.11.A, representing a function at a point as a Taylor polynomial, and AP Calc 10.11.B, using that polynomial to approximate function values near the center. You literally cannot write a Taylor polynomial without computing or reading off nth derivatives, because every coefficient is f^(n)(a)/n!. The exam also loves running this in reverse. Given a Taylor polynomial, you should be able to extract f^(n)(a) from a coefficient by multiplying by n!. That two-way skill is one of the most reliably tested moves in all of Unit 10.

## Connections

### Taylor Polynomial Approximations (Unit 10)

Every coefficient in a [Taylor polynomial](/ap-calc/unit-10-infinite-sequences-and-series-bc-only/finding-taylor-or-maclaurin-series-for-function/study-guide/aKEvYorayYkUSTp1eCXv "fv-autolink") is an nth derivative in disguise. The term for degree n is f^(n)(a)/n! times (x - a)^n, so building or decoding a Taylor polynomial is really just bookkeeping with nth derivatives.

### [Tangent Line Approximation (Unit 4)](/ap-calc/key-terms/tangent-line-approximation)

The [tangent line](/ap-calc/key-terms/tangent-line "fv-autolink") is the degree-1 Taylor polynomial. It uses only f(a) and f'(a), the zeroth and first derivatives. Higher nth derivatives are how you upgrade that line into a curve that matches f more closely. Same idea, more derivatives.

### Second Derivatives and Concavity (Units 4-5)

You already know the [first derivative](/ap-calc/key-terms/first-derivative "fv-autolink") gives slope and the second gives concavity. The nth derivative just continues that ladder. Each new derivative records a finer detail of the function's shape, which is exactly the information Taylor polynomials stack up term by term.

## On the AP Exam

On the BC exam, nth derivatives show up almost entirely inside Taylor polynomial problems. Multiple-choice stems ask things like what f^(n)(a) represents in the Taylor formula, or give you a series and ask for a specific derivative value at the center. The classic move: if the x^3 term of a Taylor polynomial centered at a is c(x - a)^3, then f^(3)(a) = 3! · c = 6c. Forgetting to multiply by n! is the most common point-loser. Free-response Taylor questions routinely hand you a table of derivative values f(a), f'(a), f''(a), f'''(a) and ask you to write the polynomial, so practice translating between derivative values and coefficients fluently in both directions.

## nth derivative vs f^n(x) (the nth power of f)

The parentheses matter. f^(n)(x) means differentiate f n times, while f^n(x) usually means raise f(x) to the nth power, like sin²(x). Reading f^(3)(x) as a cube instead of a third derivative will wreck a Taylor coefficient. Also remember n counts differentiations, so f^(0)(x) is the original function, not zero.

## Key Takeaways

- The nth derivative f^(n)(x) means you differentiate the function n times, and f^(0)(x) is just the original function itself.
- In a Taylor polynomial centered at x = a, the coefficient of the nth-degree term is f^(n)(a) divided by n!.
- To recover a derivative value from a Taylor polynomial, take the coefficient of the (x - a)^n term and multiply it by n!.
- The tangent line approximation from Unit 4 is just the degree-1 case, using only the function value and the first derivative.
- As the degree of a Taylor polynomial increases, matching more nth derivatives at the center makes the polynomial approach the original function over some interval.
- This is BC-only material from Unit 10, so AB students won't see Taylor polynomial questions on their exam.

## FAQs

### What is the nth derivative in AP Calc?

It's the function you get after differentiating f a total of n times, written f^(n)(x). On the BC exam it matters most in Topic 10.11, where f^(n)(a)/n! is the coefficient of the nth-degree term of a Taylor polynomial centered at x = a.

### Is f^(n)(x) the same as f(x) raised to the nth power?

No. The parentheses around the n signal differentiation, not exponentiation. f^(3)(x) is the third derivative, while f^3(x) typically means f(x) cubed. Mixing these up is one of the fastest ways to botch a Taylor series problem.

### What does f^(n)(a) represent in the Taylor series formula?

It's the nth derivative of f evaluated at the center x = a. Divide it by n! and you get the coefficient of the (x - a)^n term, which is exactly how the CED defines Taylor polynomial coefficients in Topic 10.11.

### How is the nth derivative different from a tangent line approximation?

A tangent line approximation only uses the first derivative, so it's the degree-1 Taylor polynomial. Higher nth derivatives let you build degree-2, degree-3, and higher polynomials that match the function's curvature, not just its slope.

### Do I need nth derivatives on the AP Calc AB exam?

Not in this Taylor polynomial sense. Unit 10 is BC-only, so Taylor polynomial coefficients and the f^(n)(a)/n! formula only appear on the BC exam. AB still uses second derivatives for concavity, but not the general nth-derivative machinery.

## Related Study Guides

- [10.11 Finding Taylor Polynomial Approximations of Functions](/ap-calc/unit-10-infinite-sequences-and-series-bc-only/finding-taylor-polynomial-approximations-functions/study-guide/LszguYzKz0M6GdqTRSr6)

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