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).
The nth derivative is exactly what it sounds like. Take the derivative 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 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, concavity, and so on at the center. The more derivatives you match, the better the polynomial hugs the function near x = a.
This term lives in Unit 10 (Infinite Sequences and Series, BC only), specifically Topic 10.11. It directly supports learning objective AP Calc 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.
Keep studying AP® Calculus Unit 10-infinite-sequences-and-series-bc-only
Taylor Polynomial Approximations (Unit 10)
Every coefficient in a Taylor polynomial 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)
The tangent line 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 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 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.
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.
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.
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.
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.
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.
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.
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.
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