AP Calculus AB Flashcards: Derivative Notation

What this deck covers

This deck focuses on Derivative Notation, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the limit definition of the derivative of $f(x)$?

Answer: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. Standard limit definition using difference quotient as $h$ approaches zero.

Flashcard 2: Identify the derivative of $f(x) = \text{e}^{-x}$.

Answer: $f'(x) = -\text{e}^{-x}$. Chain rule with exponential function and negative exponent.

Flashcard 3: State the derivative of $f(x)$ in prime notation.

Answer: $f'(x)$. Prime notation for first derivative of function $f$.

Flashcard 4: Find the derivative of $f(x) = 3x^2 + 5x - 4$.

Answer: $f'(x) = 6x + 5$. Apply power rule to each term separately.

Flashcard 5: How is the derivative of $y$ with respect to $x$ written using Leibniz's notation?

Answer: $\frac{dy}{dx}$. Standard Leibniz differential notation for derivatives.

Flashcard 6: What is the derivative of $\text{ln}(x)$?

Answer: $\frac{1}{x}$. Natural logarithm derivative is reciprocal function.

Flashcard 7: What does $D_x[f(x)]$ represent?

Answer: The derivative of $f(x)$ with respect to $x$. Operator notation where $D_x$ indicates differentiation with respect to $x$.

Flashcard 8: What is the derivative of $\text{csc}(x)$?

Answer: $-\text{csc}(x)\text{cot}(x)$. Derivative of cosecant is negative cosecant cotangent.

Flashcard 9: What is the derivative of a difference $f(x) - g(x)$?

Answer: $f'(x) - g'(x)$. Derivative of difference equals difference of derivatives.

Flashcard 10: What is the derivative of a sum $f(x) + g(x)$?

Answer: $f'(x) + g'(x)$. Derivative of sum equals sum of derivatives.

Flashcard 11: What is the derivative of a constant function $c$?

Answer: $0$. Constants have zero rate of change.

Flashcard 12: What is the derivative of $\text{log}_a(x)$?

Answer: $\frac{1}{x\text{ln}(a)}$. Logarithm base $a$ derivative involves natural log of base.

Flashcard 13: What is the derivative of a product $f(x)g(x)$?

Answer: $f'(x)g(x) + f(x)g'(x)$. Product rule for differentiating products of functions.

Flashcard 14: State the derivative of $f(x) = x\text{ln}(x)$.

Answer: $f'(x) = 1 + \text{ln}(x)$. Product rule applied to $x$ and $\ln(x)$.

Flashcard 15: What does $f'(x)$ represent graphically?

Answer: The slope of the tangent line to $f(x)$ at $x$. Derivative gives instantaneous rate of change at each point.

Flashcard 16: What is the derivative of a quotient $\frac{f(x)}{g(x)}$?

Answer: $\frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$. Quotient rule for differentiating ratios of functions.

Flashcard 17: What is the derivative of $f(x) = 3^x$?

Answer: $f'(x) = 3^x\text{ln}(3)$. Exponential with base $a$ involves $\ln(a)$ factor.

Flashcard 18: What is the definition of the derivative of a function at a point $x=a$?

Answer: $f'(a) = \frac{d}{dx}f(x) \bigg|_{x=a}$. Notation shows derivative of $f$ evaluated at point $a$.

Flashcard 19: Differentiate $f(x) = \frac{x^2}{x+1}$.

Answer: $f'(x) = \frac{x(x+2)}{(x+1)^2}$. Apply quotient rule to rational function.

Flashcard 20: State the Power Rule for derivatives.

Answer: $\frac{d}{dx}x^n = nx^{n-1}$. Fundamental derivative rule for power functions.

Flashcard 21: Find the derivative of $f(x) = \text{e}^{2x}$.

Answer: $f'(x) = 2\text{e}^{2x}$. Chain rule: derivative of outer times derivative of inner.

Flashcard 22: What is the Chain Rule in derivative notation?

Answer: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Formula for differentiating composite functions.

Flashcard 23: Differentiate $f(x) = 5\text{sin}(x)$.

Answer: $f'(x) = 5\text{cos}(x)$. Constant factor rule: $\frac{d}{dx}[cf(x)] = cf'(x)$.

Flashcard 24: Find the derivative of $f(x) = \text{ln}(x^2)$.

Answer: $f'(x) = \frac{2}{x}$. Use property $\ln(x^2) = 2\ln(x)$ then differentiate.

Flashcard 25: What is the derivative of $f(x) = \text{sin}^2(x)$?

Answer: $f'(x) = 2\text{sin}(x)\text{cos}(x)$. Chain rule with power of sine function.

Flashcard 26: Identify the derivative of $\frac{1}{x}$.

Answer: $-\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply power rule.

Flashcard 27: What is the derivative of $\text{sin}(x)$?

Answer: $\text{cos}(x)$. Derivative of sine is cosine.

Flashcard 28: Identify the derivative notation for $y$ with respect to $x$.

Answer: $\frac{dy}{dx}$. Leibniz notation for derivative of $y$ with respect to $x$.

Flashcard 29: What is the derivative of $\text{sec}(x)$?

Answer: $\text{sec}(x)\text{tan}(x)$. Derivative of secant involves secant times tangent.

Flashcard 30: What is the derivative of $\text{sec}(x)$?

Answer: $\text{sec}(x)\text{tan}(x)$. Derivative of secant involves secant times tangent.