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: 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 2: 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 3: 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 4: 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 5: 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 6: What is the limit definition of the derivative of $f(x)$?

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

Flashcard 7: What is the derivative of $\text{csc}(x)$?

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

Flashcard 8: 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 9: Find $f'(x)$ for $f(x) = \text{cos}(2x)$.

Answer: $f'(x) = -2\text{sin}(2x)$. Chain rule with cosine and linear inner function.

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

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

Flashcard 13: State the Power Rule for derivatives.

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

Flashcard 14: 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 15: State the Power Rule for derivatives.

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

Flashcard 16: 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 17: What is the derivative of $e^x$?

Answer: $e^x$. Exponential function $e^x$ is its own derivative.

Flashcard 18: 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 19: State the derivative of $f(x)$ in prime notation.

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

Flashcard 20: 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 21: What is the derivative of $\sin(x)$?

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

Flashcard 22: Identify the derivative of $f(x) = \text{e}^{-x}$.

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

Flashcard 23: What is the derivative of $f(x) = 3^x$?

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

Flashcard 24: 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 25: What is the derivative of a difference $f(x) - g(x)$?

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

Flashcard 26: 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 27: What is the derivative of a sum $f(x) + g(x)$?

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

Flashcard 28: 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 29: What is the derivative of $\text{cos}(x)$?

Answer: $- ext{sin}(x)$. Derivative of cosine is negative sine.

Flashcard 30: Differentiate $f(x) = x^5 - 4x^3 + 2$.

Answer: $f'(x) = 5x^4 - 12x^2$. Apply power rule to each polynomial term.

Flashcard 31: 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 32: What is the derivative of $\text{csc}(x)$?

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

Flashcard 33: Find $f'(x)$ for $f(x) = \text{cos}(2x)$.

Answer: $f'(x) = -2\text{sin}(2x)$. Chain rule with cosine and linear inner function.

Flashcard 34: State the derivative of $f(x)$ in prime notation.

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

Flashcard 35: What is the derivative of $\text{cot}(x)$?

Answer: $-\text{csc}^2(x)$. Derivative of cotangent is negative cosecant squared.

Flashcard 36: 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 37: 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 38: What is the derivative of $\text{cot}(x)$?

Answer: $-\text{csc}^2(x)$. Derivative of cotangent is negative cosecant squared.

Flashcard 39: 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 40: 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 41: Find the derivative of $g(x) = \frac{2}{x}$.

Answer: $g'(x) = -\frac{2}{x^2}$. Rewrite as $2x^{-1}$ and use power rule.

Flashcard 42: Determine the derivative of $h(x) = x^3 + 4x$.

Answer: $h'(x) = 3x^2 + 4$. Power rule applied to each term.

Flashcard 43: What is the derivative of $\text{ln}(x)$?

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

Flashcard 44: 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 45: 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 46: What is the derivative of a difference $f(x) - g(x)$?

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

Flashcard 47: What is the derivative of $\log_a(x)$?

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

Flashcard 48: 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 49: What is the derivative of $\text{tan}(x)$?

Answer: $\text{sec}^2(x)$. Derivative of tangent is secant squared.

Flashcard 50: 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 51: 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 52: Find the derivative of $f(x) = 3x^2 + 5x - 4$.

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

Flashcard 53: 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 54: What is the derivative of $\text{ln}(x)$?

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

Flashcard 55: 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 56: 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 57: What is the derivative of $\text{sec}(x)$?

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

Flashcard 58: Identify the derivative of $f(x) = \text{e}^{-x}$.

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

Flashcard 59: What is the derivative of $x^n$ where $n$ is a constant?

Answer: $nx^{n-1}$. Power rule: bring down exponent, reduce power by one.

Flashcard 60: Differentiate $f(x) = 5\sin(x)$.

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

Flashcard 61: 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 62: What is the derivative of $\text{sin}(x)$?

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

Flashcard 63: 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 64: 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 65: What is the derivative of a constant function $c$?

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

Flashcard 66: Identify the derivative of $\frac{1}{x}$.

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

Flashcard 67: What is the derivative of a constant function $c$?

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

Flashcard 68: Find the derivative of $f(x) = 3x^2 + 5x - 4$.

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

Flashcard 69: Identify the derivative of $\frac{1}{x}$.

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