AP Calculus AB Practice Quiz: Calculating Higher-Order Derivatives
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Test your understanding with short quizzes. This quiz has 9 questions to check your progress.
Question 1 of 9
All Questions (9)
A) f'(x)²
B) f²(x)
C) f''(x)
D) 2f'(x)
Correct Answer: C
Based on the provided content, notations for the second derivative include d²y/dx², f''(x), and y''. The notation f''(x) correctly represents the second derivative, which is found by differentiating the first derivative, f'(x).
A) 12x³ + 10x
B) 36x² + 10
C) 72x
D) 12x² + 5
Correct Answer: B
To find the second derivative, first find the first derivative: f'(x) = d/dx(3x⁴ + 5x²) = 12x³ + 10x. Then, differentiate f'(x) to get the second derivative: f''(x) = d/dx(12x³ + 10x) = 36x² + 10.
A) Differentiating the function f(x) four times.
B) Taking the fourth power of the first derivative, (f'(x))⁴.
C) Finding the derivative of the function f(x) and multiplying it by four.
D) Integrating the function f(x) four times.
Correct Answer: A
The content states that higher-order derivatives are produced by repeating the process of differentiation. The second derivative is the derivative of the first, the third is the derivative of the second, and so on. Therefore, the fourth derivative is found by differentiating the function a total of four times.
A) 5x⁴ - 6x² + 7
B) 20x³ - 12x
C) 60x² - 12
D) 120x
Correct Answer: C
The notation d³y/dx³ asks for the third derivative. We differentiate three times: 1. dy/dx = 5x⁴ - 6x² + 7 2. d²y/dx² = 20x³ - 12x 3. d³y/dx³ = 60x² - 12
A) f(x)
B) f'(x)
C) f''(x)
D) f⁽⁴⁾(x)
Correct Answer: C
The content states that differentiating f' produces f''. Repeating this process, differentiating the second derivative, f''(x), produces the third derivative, f'''(x).
A) 5eˣ
B) e⁵ˣ
C) eˣ
D) 0
Correct Answer: C
The notation f⁽⁵⁾(x) represents the fifth derivative of f(x). The derivative of eˣ is eˣ. Therefore, every higher-order derivative of eˣ is also eˣ.
A) 48
B) 24x - 54
C) 0
D) 2
Correct Answer: C
We find the derivatives sequentially: f'(x) = 8x³ - 27x² + 1 f''(x) = 24x² - 54x f'''(x) = 48x - 54 f⁽⁴⁾(x) = 48 f⁽⁵⁾(x) = 0 The derivative of a constant (48) is zero.
A) -1/x²
B) 2/x³
C) -6/x⁴
D) 1/x
Correct Answer: B
First, rewrite y as x⁻¹. The first derivative, dy/dx, is -1 * x⁻² = -1/x². The second derivative, d²y/dx², is the derivative of -1x⁻², which is (-1)(-2)x⁻³ = 2x⁻³ = 2/x³. Option A is the first derivative.
A) sin(x)
B) -sin(x)
C) cos(x)
D) -cos(x)
Correct Answer: C
The derivatives of cos(x) follow a cycle of four: y' = -sin(x) y'' = -cos(x) y''' = sin(x) y⁽⁴⁾ = cos(x) The pattern repeats every four derivatives. Since 20 is a multiple of 4 (20 / 4 = 5), the 20th derivative will be the same as the 4th derivative, which is cos(x).