AP Calculus AB Flashcards: Sketching Graphs Of Functions And Derivatives

What this deck covers

This deck focuses on Sketching Graphs Of Functions And Derivatives, 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 $f(x) = \text{tan}(x)$?

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

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

Answer: $f'(x) = a^x \text{ln}(a)$. General exponential derivative includes $\ln(a)$.

Flashcard 3: Differentiate $f(x) = \frac{1}{x}$.

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

Flashcard 4: State the Power Rule for differentiation.

Answer: $\frac{d}{dx}x^n = nx^{n-1}$. Multiply by exponent, reduce exponent by 1.

Flashcard 5: Calculate $f'(x)$ for $f(x) = \text{cos}(x)$.

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

Flashcard 6: What is the second derivative of $f(x) = 3x^4$?

Answer: $f''(x) = 36x^2$. Apply power rule twice: $f'(x) = 12x^3$, then again.

Flashcard 7: What is $f'(x)$ for $f(x) = \text{sin}(x)$?

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

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

Answer: $f'(x) = \frac{1}{x}$. The natural logarithm's derivative is $\frac{1}{x}$.

Flashcard 9: Calculate $f'(x)$ for $f(x) = \text{arccos}(x)$.

Answer: $f'(x) = -\frac{1}{\text{√}(1-x^2)}$. Negative of arcsine derivative.

Flashcard 10: Identify the critical points of $f(x) = x^3 - 3x$.

Answer: $x = 0, x = \text{±}\frac{\text{√}3}{3}$. Set $f'(x) = 3x^2 - 3 = 0$ and solve for x.

Flashcard 11: Calculate $f'(x)$ for $f(x) = \frac{1}{2}x^4$.

Answer: $f'(x) = 2x^3$. Apply power rule: $4 \cdot \frac{1}{2} \cdot x^3 = 2x^3$.

Flashcard 12: Find $f'(x)$ for $f(x) = \text{csc}(x)$.

Answer: $f'(x) = -\text{csc}(x)\text{cot}(x)$. Derivative involves negative cotangent cosecant.

Flashcard 13: What is the derivative of $f(x) = \arcsin(x)$?

Answer: $f'(x) = \frac{1}{\sqrt{1 - x^2}}$. Inverse trig derivative with radical denominator.

Flashcard 14: What is the derivative of $f(x) = x^2$?

Answer: $f'(x) = 2x$. Power rule: bring down exponent, subtract 1.

Flashcard 15: What is $f'(x)$ for $f(x) = \text{arctan}(x)$?

Answer: $f'(x) = \frac{1}{1+x^2}$. Inverse tangent derivative with squared denominator.

Flashcard 16: Find $f'(x)$ for $f(x) = \text{arccot}(x)$.

Answer: $f'(x) = -\frac{1}{1+x^2}$. Negative of arctangent derivative.

Flashcard 17: What indicates a local maximum in $f'(x)$?

Answer: Change from positive to negative. Derivative sign changes from + to - at local max.

Flashcard 18: What does $f'(x) = 0$ indicate about $f(x)$?

Answer: Potential extrema. Horizontal tangent lines occur at critical points.

Flashcard 19: State the Quotient Rule for differentiation.

Answer: $\frac{d}{dx}[\frac{u}{v}] = \frac{u'v - uv'}{v^2}$. Low d-high minus high d-low over low squared.

Flashcard 20: What is the Product Rule for differentiation?

Answer: $\frac{d}{dx}[uv] = u'v + uv'$. Sum of each function times the other's derivative.

Flashcard 21: Find $f'(x)$ for $f(x) = \text{csc}(x)$.

Answer: $f'(x) = -\text{csc}(x)\text{cot}(x)$. Derivative involves negative cotangent cosecant.

Flashcard 22: Differentiate $f(x) = 7x^3 - 2x$.

Answer: $f'(x) = 21x^2 - 2$. Apply power rule to each term separately.

Flashcard 23: Find $f'(x)$ for $f(x) = \text{sec}(x)$.

Answer: $f'(x) = \text{sec}(x)\text{tan}(x)$. Product of secant and tangent functions.

Flashcard 24: What is the Product Rule for differentiation?

Answer: $\frac{d}{dx}[uv] = u'v + uv'$. Sum of each function times the other's derivative.

Flashcard 25: Identify the intervals where $f(x) = x^2 - 4$ is increasing.

Answer: $x > 0$. $f'(x) = 2x > 0$ when $x > 0$.

Flashcard 26: State the Chain Rule for differentiation.

Answer: $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$. Differentiate outer function times inner derivative.

Flashcard 27: State the Chain Rule for differentiation.

Answer: $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$. Differentiate outer function times inner derivative.

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

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

Flashcard 29: Differentiate $f(x) = 7x^3 - 2x$.

Answer: $f'(x) = 21x^2 - 2$. Apply power rule to each term separately.

Flashcard 30: What is $f'(x)$ for $f(x) = x^5$?

Answer: $f'(x) = 5x^4$. Power rule: bring down 5, subtract 1 from exponent.

Flashcard 31: What is the inverse function derivative formula?

Answer: $[f^{-1}]'(x) = \frac{1}{f'(f^{-1}(x))}$. Reciprocal of derivative at corresponding point.

Flashcard 32: Find the critical points of $f(x) = x^2 - 6x + 8$.

Answer: $x = 3$. Set $f'(x) = 2x - 6 = 0$ and solve.

Flashcard 33: What indicates a local maximum in $f'(x)$?

Answer: Change from positive to negative. Derivative sign changes from + to - at local max.

Flashcard 34: Which test identifies concavity?

Answer: Second Derivative Test. Examines sign of $f''(x)$ to determine concavity.

Flashcard 35: What is $f'(x)$ for $f(x) = \text{arctan}(x)$?

Answer: $f'(x) = \frac{1}{1+x^2}$. Inverse tangent derivative with squared denominator.

Flashcard 36: Differentiate $f(x) = \frac{1}{x}$.

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

Flashcard 37: What is $f'(x)$ for $f(x) = \text{sin}(x)$?

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

Flashcard 38: State the Quotient Rule for differentiation.

Answer: $\frac{d}{dx}[\frac{u}{v}] = \frac{u'v - uv'}{v^2}$. Low d-high minus high d-low over low squared.

Flashcard 39: Find $f'(x)$ for $f(x) = \text{arccot}(x)$.

Answer: $f'(x) = -\frac{1}{1+x^2}$. Negative of arctangent derivative.

Flashcard 40: Which test identifies concavity?

Answer: Second Derivative Test. Examines sign of $f''(x)$ to determine concavity.

Flashcard 41: Identify the concavity of $f(x) = x^4$.

Answer: Concave up for all $x$. $f''(x) = 12x^2 \geq 0$ for all real x.

Flashcard 42: Identify the intervals where $f(x) = x^2 - 4$ is increasing.

Answer: $x > 0$. $f'(x) = 2x > 0$ when $x > 0$.

Flashcard 43: Which test uses $f''(x)$ to find extrema?

Answer: Second Derivative Test. Uses second derivative to classify critical points.

Flashcard 44: What is the derivative of $f(x) = \text{cot}(x)$?

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

Flashcard 45: What is the second derivative of $f(x) = 3x^4$?

Answer: $f''(x) = 36x^2$. Apply power rule twice: $f'(x) = 12x^3$, then again.

Flashcard 46: What is the derivative of a constant $c$?

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

Flashcard 47: Calculate $f'(x)$ for $f(x) = \text{cos}(x)$.

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

Flashcard 48: Find the derivative of $f(x) = a^x$.

Answer: $f'(x) = a^x \text{ln}(a)$. General exponential derivative includes $\ln(a)$.

Flashcard 49: Find $f'(x)$ for $f(x) = \text{ln}(x)$.

Answer: $f'(x) = \frac{1}{x}$. The natural logarithm's derivative is $\frac{1}{x}$.

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

Answer: $f'(x) = \frac{1}{\text{√}(1-x^2)}$. Inverse trig derivative with radical denominator.

Flashcard 51: Find the critical points of $f(x) = x^2 - 6x + 8$.

Answer: $x = 3$. Set $f'(x) = 2x - 6 = 0$ and solve.

Flashcard 52: What does $f'(x) = 0$ indicate about $f(x)$?

Answer: Potential extrema. Horizontal tangent lines occur at critical points.

Flashcard 53: State the Power Rule for differentiation.

Answer: $\frac{d}{dx}x^n = nx^{n-1}$. Multiply by exponent, reduce exponent by 1.

Flashcard 54: Identify the concavity of $f(x) = x^4$.

Answer: Concave up for all $x$. $f''(x) = 12x^2 \geq 0$ for all real x.

Flashcard 55: What is the inverse function derivative formula?

Answer: $[f^{-1}]'(x) = \frac{1}{f'(f^{-1}(x))}$. Reciprocal of derivative at corresponding point.

Flashcard 56: What is the derivative of $f(x) = e^x$?

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

Flashcard 57: What is the derivative of $f(x) = e^x$?

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

Flashcard 58: What is the derivative of a constant $c$?

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

Flashcard 59: Find $f'(x)$ for $f(x) = \text{sec}(x)$.

Answer: $f'(x) = \text{sec}(x)\text{tan}(x)$. Product of secant and tangent functions.

Flashcard 60: Calculate $f'(x)$ for $f(x) = \arccos(x)$.

Answer: f'(x) = -\frac{1}{\sqrt{1-x^2)}. Negative of arcsine derivative.

Flashcard 61: Identify the critical points of $f(x) = x^3 - 3x$.

Answer: $x = 0, x = \text{±}\frac{\text{√}3}{3}$. Set $f'(x) = 3x^2 - 3 = 0$ and solve for x.

Flashcard 62: Calculate $f'(x)$ for $f(x) = \frac{1}{2}x^4$.

Answer: $f'(x) = 2x^3$. Apply power rule: $4 \cdot \frac{1}{2} \cdot x^3 = 2x^3$.

Flashcard 63: Which test uses $f''(x)$ to find extrema?

Answer: Second Derivative Test. Uses second derivative to classify critical points.

Flashcard 64: What is the derivative of $f(x) = \text{tan}(x)$?

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

Flashcard 65: What is the derivative of $f(x) = x^2$?

Answer: $f'(x) = 2x$. Power rule: bring down exponent, subtract 1.

Flashcard 66: What is $f'(x)$ for $f(x) = x^5$?

Answer: $f'(x) = 5x^4$. Power rule: bring down 5, subtract 1 from exponent.