AP Calculus AB Flashcards: Introducing Calculus

What this deck covers

This deck focuses on Introducing Calculus, 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 derivative of $f(x) = a^x$, where $a > 0$?

Answer: $f'(x) = a^x \text{ln}(a)$. For exponential with base $a$, multiply by $\ln(a)$.

Flashcard 2: What is the Quotient Rule for differentiation?

Answer: If $\frac{u}{v}$, then $\frac{vu' - uv'}{v^2}$. For quotients: bottom times top's derivative minus top times bottom's derivative, all over bottom squared.

Flashcard 3: State the limit definition of a derivative.

Answer: $f'(x) = \frac{d}{dx}f(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. This is the formal limit definition using the difference quotient.

Flashcard 4: Identify the derivative of $f(x) = \frac{1}{x}$.

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

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

Answer: $f'(x) = -\frac{1}{|x|\text{sqrt}(x^2-1)}$. Similar to arcsec but with negative sign.

Flashcard 6: Apply the Product Rule to $f(x) = x^2 \text{sin}(x)$.

Answer: $f'(x) = 2x\text{sin}(x) + x^2\text{cos}(x)$. Apply product rule: $u = x^2$, $v = \sin(x)$.

Flashcard 7: What is the derivative of a constant function $f(x) = c$?

Answer: $f'(x) = 0$. Constants have no change, so their rate of change is zero.

Flashcard 8: Apply the Chain Rule to $f(x) = (3x^2 + 2)^5$.

Answer: $f'(x) = 5(3x^2 + 2)^4 \times 6x$. Outer function derivative is $5(3x^2 + 2)^4$, inner is $6x$.

Flashcard 9: Differentiate $f(x) = 7x + 4$.

Answer: $f'(x) = 7$. Linear term becomes its coefficient; constant disappears.

Flashcard 10: State the limit definition of a derivative.

Answer: $f'(x) = \frac{d}{dx}f(x) = \text{lim}_{h \to 0} \frac{f(x+h) - f(x)}{h}$. This is the formal limit definition using the difference quotient.

Flashcard 11: Identify the derivative of $f(x) = x^2$.

Answer: $f'(x) = 2x$. Using the power rule: bring down the exponent and subtract 1.

Flashcard 12: Differentiate $f(x) = \text{e}^{-x^2}$ using the Chain Rule.

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

Flashcard 13: Apply the Product Rule to $f(x) = x^2 \text{sin}(x)$.

Answer: $f'(x) = 2x\text{sin}(x) + x^2\text{cos}(x)$. Apply product rule: $u = x^2$, $v = \sin(x)$.

Flashcard 14: Differentiate $f(x) = x^3 - 5x + 4$.

Answer: $f'(x) = 3x^2 - 5$. Apply power rule term by term: $3x^2 - 5 + 0$.

Flashcard 15: Differentiate $f(x) = \text{arctan}(x)$.

Answer: $f'(x) = \frac{1}{1+x^2}$. This is the standard derivative formula for inverse tangent.

Flashcard 16: What is the derivative of $f(x) = a^x$, where $a > 0$?

Answer: $f'(x) = a^x \text{ln}(a)$. For exponential with base $a$, multiply by $\ln(a)$.

Flashcard 17: What is the Chain Rule for differentiation?

Answer: If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x))g'(x)$. This rule handles composite functions: outer derivative times inner.

Flashcard 18: Identify the derivative of $f(x) = \text{arccot}(x)$.

Answer: $f'(x) = -\frac{1}{1+x^2}$. Similar to arctan but with negative sign.

Flashcard 19: Differentiate $f(x) = \text{ln}(3x)$ using the Chain Rule.

Answer: $f'(x) = \frac{3}{3x} = \frac{1}{x}$. Chain rule: derivative of $\ln(u)$ is $\frac{u'}{u}$.

Flashcard 20: Differentiate $f(x) = \text{sin}(x)$.

Answer: $f'(x) = \text{cos}(x)$. The derivative of sine is cosine.

Flashcard 21: Apply the Power Rule to $f(x) = x^5$.

Answer: $f'(x) = 5x^4$. Using power rule: $5x^{5-1} = 5x^4$.

Flashcard 22: Differentiate $f(x) = \text{tan}(x)$.

Answer: $f'(x) = \text{sec}^2(x)$. The derivative of tangent is secant squared.

Flashcard 23: What is the derivative of $f(x) = \text{cot}(x)$?

Answer: $f'(x) = -\text{csc}^2(x)$. The derivative of cotangent is negative cosecant squared.

Flashcard 24: Differentiate $f(x) = 7x + 4$.

Answer: $f'(x) = 7$. Linear term becomes its coefficient; constant disappears.

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

Answer: $f'(x) = \frac{1}{|x| \sqrt{x^2-1}}$. The derivative includes absolute value due to domain restrictions.

Flashcard 26: Differentiate $f(x) = \text{arccsc}(x)$.

Answer: $f'(x) = -\frac{1}{|x|\text{sqrt}(x^2-1)}$. Similar to arcsec but with negative sign.

Flashcard 27: Find the derivative of $f(x) = 3x^3$.

Answer: $f'(x) = 9x^2$. Apply power rule to $x^3$: $3 \cdot 3x^2 = 9x^2$.

Flashcard 28: What is the derivative of $f(x) = \text{ln}(x)$?

Answer: $f'(x) = \frac{1}{x}$. This is a fundamental derivative of logarithmic functions.

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

Answer: $f'(x) = -\text{sin}(x)$. The derivative of cosine is negative sine.

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

Answer: $f'(x) = \text{sec}(x)\text{tan}(x)$. This follows from the chain rule applied to $\sec(x) = \frac{1}{\cos(x)}$.

Flashcard 31: What is the definition of a derivative?

Answer: The derivative is the instantaneous rate of change of a function. This is the fundamental concept of calculus.

Flashcard 32: What is the definition of a derivative?

Answer: The derivative is the instantaneous rate of change of a function. This is the fundamental concept of calculus.

Flashcard 33: What is the derivative of $f(x) = \text{arcsec}(x)$?

Answer: $f'(x) = \frac{1}{|x|\text{sqrt}(x^2-1)}$. The derivative includes absolute value due to domain restrictions.

Flashcard 34: Identify the derivative of $f(x) = \text{e}^{3x}$.

Answer: $f'(x) = 3\text{e}^{3x}$. Chain rule: $e^{3x}$ times derivative of $3x$.

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

Answer: $f'(x) = -\frac{1}{\text{sqrt}(1-x^2)}$. Similar to arcsin but with negative sign.

Flashcard 36: What does the derivative tell us about a function at a point?

Answer: The slope of the tangent line to the function at that point. The derivative measures the steepness of the curve at any point.

Flashcard 37: Differentiate $f(x) = x^3 - 5x + 4$.

Answer: $f'(x) = 3x^2 - 5$. Apply power rule term by term: $3x^2 - 5 + 0$.

Flashcard 38: Identify the derivative of $f(x) = \text{csc}(x)$.

Answer: $f'(x) = -\text{csc}(x)\text{cot}(x)$. This follows from the chain rule applied to $\csc(x) = \frac{1}{\sin(x)}$.

Flashcard 39: Identify the derivative of $f(x) = \frac{1}{x}$.

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

Flashcard 40: What is the derivative of a constant function $f(x) = c$?

Answer: $f'(x) = 0$. Constants have no change, so their rate of change is zero.

Flashcard 41: Apply the Power Rule to $f(x) = x^5$.

Answer: $f'(x) = 5x^4$. Using power rule: $5x^{5-1} = 5x^4$.

Flashcard 42: What is the derivative of $f(x) = \text{arccos}(x)$?

Answer: $f'(x) = -\frac{1}{\text{sqrt}(1-x^2)}$. Similar to arcsin but with negative sign.

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

Answer: $f'(x) = \text{sec}(x)\text{tan}(x)$. This follows from the chain rule applied to $\sec(x) = \frac{1}{\cos(x)}$.

Flashcard 44: What is the derivative of $f(x) = e^x$?

Answer: $f'(x) = e^x$. The exponential function is its own derivative.

Flashcard 45: What is the Chain Rule for differentiation?

Answer: If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x))g'(x)$. This rule handles composite functions: outer derivative times inner.

Flashcard 46: What is the derivative of $f(x) = \text{log}_a(x)$?

Answer: $f'(x) = \frac{1}{x \text{ln}(a)}$. For logarithm base $a$, include the factor $\frac{1}{\ln(a)}$.

Flashcard 47: Apply the Chain Rule to $f(x) = (3x^2 + 2)^5$.

Answer: $f'(x) = 5(3x^2 + 2)^4 \times 6x$. Outer function derivative is $5(3x^2 + 2)^4$, inner is $6x$.

Flashcard 48: Identify the derivative of $f(x) = \text{arccot}(x)$.

Answer: $f'(x) = -\frac{1}{1+x^2}$. Similar to arctan but with negative sign.

Flashcard 49: What is the Quotient Rule for differentiation?

Answer: If $\frac{u}{v}$, then $\frac{vu' - uv'}{v^2}$.. For quotients: bottom times top's derivative minus top times bottom's derivative, all over bottom squared.

Flashcard 50: What is the derivative of $f(x) = \text{cos}(x)$?

Answer: $f'(x) = -\text{sin}(x)$. The derivative of cosine is negative sine.

Flashcard 51: Differentiate $f(x) = \text{arcsin}(x)$.

Answer: $f'(x) = \frac{1}{\text{sqrt}(1-x^2)}$. This is the standard derivative formula for inverse sine.

Flashcard 52: Differentiate $f(x) = \text{arctan}(x)$.

Answer: $f'(x) = \frac{1}{1+x^2}$. This is the standard derivative formula for inverse tangent.

Flashcard 53: Differentiate $f(x) = \text{arcsin}(x)$.

Answer: $f'(x) = \frac{1}{\text{sqrt}(1-x^2)}$. This is the standard derivative formula for inverse sine.

Flashcard 54: Differentiate $f(x) = \text{ln}(3x)$ using the Chain Rule.

Answer: $f'(x) = \frac{3}{3x} = \frac{1}{x}$. Chain rule: derivative of $\ln(u)$ is $\frac{u'}{u}$.

Flashcard 55: What does the derivative tell us about a function at a point?

Answer: The slope of the tangent line to the function at that point. The derivative measures the steepness of the curve at any point.

Flashcard 56: What is the Power Rule for differentiation?

Answer: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.. This is the fundamental differentiation rule for polynomial terms.

Flashcard 57: Find the derivative of $f(x) = 3x^3$.

Answer: $f'(x) = 9x^2$. Apply power rule to $x^3$: $3 \cdot 3x^2 = 9x^2$.

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

Answer: $f'(x) = -\text{csc}(x)\text{cot}(x)$. This follows from the chain rule applied to $\csc(x) = \frac{1}{\sin(x)}$.

Flashcard 59: What is the derivative of $f(x) = \text{log}_a(x)$?

Answer: $f'(x) = \frac{1}{x \ln(a)}$. For logarithm base $a$, include the factor $\frac{1}{\ln(a)}$.

Flashcard 60: What is the Product Rule for differentiation?

Answer: If $u(x)v(x)$, then $uv' + vu'$.. For products: first times derivative of second plus second times derivative of first.

Flashcard 61: Differentiate $f(x) = \text{tan}(x)$.

Answer: $f'(x) = \text{sec}^2(x)$. The derivative of tangent is secant squared.

Flashcard 62: Identify the derivative of $f(x) = \text{e}^{3x}$.

Answer: $f'(x) = 3\text{e}^{3x}$. Chain rule: $e^{3x}$ times derivative of $3x$.

Flashcard 63: Differentiate $f(x) = \text{e}^{-x^2}$ using the Chain Rule.

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

Flashcard 64: What is the Power Rule for differentiation?

Answer: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.. This is the fundamental differentiation rule for polynomial terms.

Flashcard 65: Differentiate $f(x) = \text{sin}(x)$.

Answer: $f'(x) = \text{cos}(x)$. The derivative of sine is cosine.

Flashcard 66: Identify the derivative of $f(x) = x^2$.

Answer: $f'(x) = 2x$. Using the power rule: bring down the exponent and subtract 1.