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 $f(x) = \text{ln}(x)$?

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

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

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

Flashcard 3: 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 4: 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 5: 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 6: Differentiate $f(x) = \text{sin}(x)$.

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

Flashcard 7: 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 8: 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 9: Differentiate $f(x) = x^3 - 5x + 4$.

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

Flashcard 10: What is the derivative of $f(x) = \text{cos}(x)$?

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

Flashcard 11: 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 12: 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 13: Identify the derivative of $f(x) = x^2$.

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

Flashcard 14: 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 15: 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 16: 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 17: 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 18: 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 19: 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 20: Apply the Power Rule to $f(x) = x^5$.

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

Flashcard 21: 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 22: 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 23: 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 24: 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 25: 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 26: Differentiate $f(x) = 7x + 4$.

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

Flashcard 27: Differentiate $f(x) = \text{tan}(x)$.

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

Flashcard 28: 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 29: 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 30: Apply the Power Rule to $f(x) = x^5$.

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