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AP Calculus AB Unit 2: Differentiation and Fundamental Derivative Rules — practice questions

Explore 93 free Unit 2: Differentiation and Fundamental Derivative Rules questions in AP Calculus AB Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Unit 2: Differentiation and Fundamental Derivative Rules questions and exercises with answers

Showing questions 1–30 of 93.

$\dfrac{d}{dx}(x^n)$ equals:Read the answer and worked solution →$\dfrac{d}{dx}(\sin x)$ equals:Read the answer and worked solution →$\dfrac{d}{dx}(e^x)$ equals:Read the answer and worked solution →$\dfrac{d}{dx}(\ln x)$ equals:Read the answer and worked solution →The **chain rule** states $\dfrac{dy}{dx}$ for $y = f(g(x))$ is:Read the answer and worked solution →The product rule for $\dfrac{d}{dx}(uv)$ is:Read the answer and worked solution →$\dfrac{d}{dx}(\tan x)$ equals:Read the answer and worked solution →$\dfrac{d}{dx}(\sin^{-1} x)$ equals:Read the answer and worked solution →If $y = \sin(\cos x)$, then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →If $y = x^x$, then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →If $y = \log(\sin x)$, then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →$\dfrac{d}{dx}(a^x)$ equals (where $a > 0$):Read the answer and worked solution →If $x^2 + y^2 = 25$, then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →If $x = a t^2$, $y = 2 a t$ (parametric), then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →If $y = \sin x$, then $y_n = \dfrac{d^n y}{dx^n}$ at $x = 0$ for $n = 4$ equals:Read the answer and worked solution →If $y = \tan^{-1}\left(\dfrac{2x}{1-x^2}\right)$, then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →If $y = x \sin x$, then $y_2 = \dfrac{d^2y}{dx^2}$ equals:Read the answer and worked solution →If $x^y = y^x$, then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →If $y = \sin^{-1}(2x\sqrt{1-x^2})$ for $-1/\sqrt 2 < x < 1/\sqrt 2$, then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →$\dfrac{d}{dx}\left[\log(\log x)\right]$ at $x = e^e$ equals:Read the answer and worked solution →If $y = (\sin x)^x$, then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →Given $x = a(\theta - \sin\theta)$, $y = a(1 - \cos\theta)$ (cycloid). Then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →If $y = e^x \cos x$, then $\dfrac{d^2 y}{dx^2} - 2\dfrac{dy}{dx} + 2y$ equals:Read the answer and worked solution →If $f(x) = |x - 1| + |x - 3|$, then $f'(2)$ equals:Read the answer and worked solution →If $y_n$ denotes the $n$-th derivative of $y = \sin x$, then $y_n = $:Read the answer and worked solution →If $f(x) = x^3 - 3x$, the values of $x$ where $f'(x) = 0$ are:Read the answer and worked solution →$\dfrac{d}{dx}\left[\cos^{-1}\left(\dfrac{1-x^2}{1+x^2}\right)\right]$ for $x > 0$ equals:Read the answer and worked solution →Let $A = \begin{bmatrix} 5 & 5\alpha & \alpha \\ 0 & \alpha & 5\alpha \\ 0 & 0 & 5 \end{bmatrix}$. If $|A^2| = 25$, then $|\alpha|$ equalsRead the answer and worked solution →If $A = \begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix}$, then value of $\alpha$ for which $A^2 = B$, isRead the answer and worked solution →The angle $\theta$ between two planes with normal vectors $\vec{n}_1$ and $\vec{n}_2$ satisfiesRead the answer and worked solution →

Worked practice questions

$\dfrac{d}{dx}(x^n)$ equals:

  1. $x^{n-1}$
  2. $n x^n$
  3. $n x^{n-1}$
  4. $(n-1) x^n$
Show answer and solution

Correct answer: C. $n x^{n-1}$

Power rule: $\frac{d}{dx}(x^n) = n x^{n-1}$ for any real $n$.

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$\dfrac{d}{dx}(\sin x)$ equals:

  1. $-\sin x$
  2. $-\cos x$
  3. $\cos x$
  4. $\sec^2 x$
Show answer and solution

Correct answer: C. $\cos x$

Standard result: $d/dx(\sin x) = \cos x$.

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$\dfrac{d}{dx}(e^x)$ equals:

  1. $e^x$
  2. $\ln x$
  3. $x e^{x-1}$
  4. $1/x$
Show answer and solution

Correct answer: A. $e^x$

$e^x$ is its own derivative — unique property of the exponential function.

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More Mathematics topics

Unit 3B: Implicit Differentiation and Inverse FunctionsUnit 3A: Chain Rule and Composite FunctionsUnit 1A3: Function Composition and InversesUnit 1A1: Relations, Functions and GraphsUnit 1A2: Trigonometric Functions & Inverse TrigUnit 6A: Indefinite Integrals & Fundamental TheoremUnit 1B3: Asymptotic Behavior and Infinite LimitsUnit 1B1: Limit Concepts and Graphical Limits