AP Calculus BC Flashcards: Selecting Procedures For Calculating Derivatives

What this deck covers

This deck focuses on Selecting Procedures For Calculating Derivatives, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

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: Determine the derivative of $f(x) = x^3 - 4x + 6$.

Answer: $f'(x) = 3x^2 - 4$. Apply power rule term by term.

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

Answer: $f'(x) = 2x\text{sec}^2(x^2)$. Chain rule with tangent: $\sec^2(x^2) \cdot 2x$.

Flashcard 3: What is the derivative of $\text{sec}(x)$ with respect to $x$?

Answer: $\frac{d}{dx}[\text{sec}(x)] = \text{sec}(x)\text{tan}(x)$. Secant derivative involves tangent.

Flashcard 4: Identify the product rule for derivatives.

Answer: $(uv)' = u'v + uv'$. Product rule: first times derivative of second plus second times derivative of first.

Flashcard 5: Find the derivative of $f(x) = \text{e}^{x^2}$.

Answer: $f'(x) = 2x\text{e}^{x^2}$. Chain rule: $e^{x^2} \cdot 2x$.

Flashcard 6: What is the derivative of $f(x) = x^n$ with respect to $x$?

Answer: $f'(x) = nx^{n-1}$. Power rule: bring down exponent, reduce power by 1.

Flashcard 7: Find the derivative of $f(x) = \text{sec}^2(x)$.

Answer: $f'(x) = 2\text{sec}^2(x)\text{sec}(x)\text{tan}(x)$. Chain rule: $2\sec(x) \cdot \sec(x)\tan(x)$.

Flashcard 8: Calculate the derivative of $f(x) = \text{tan}(3x)$.

Answer: $f'(x) = 3\text{sec}^2(3x)$. Chain rule with tangent: $\sec^2(3x) \cdot 3$.

Flashcard 9: State the derivative of $\text{tan}(x)$ with respect to $x$.

Answer: $\frac{d}{dx}[\text{tan}(x)] = \text{sec}^2(x)$. Derivative of tangent is secant squared.

Flashcard 10: What is the derivative of $\text{sec}(x)$ with respect to $x$?

Answer: $\frac{d}{dx}[\text{sec}(x)] = \text{sec}(x)\text{tan}(x)$. Secant derivative involves tangent.

Flashcard 11: Calculate the derivative of $f(x) = \text{cos}(x^2)$.

Answer: $f'(x) = -2x\text{sin}(x^2)$. Chain rule: $-\sin(x^2) \cdot 2x$.

Flashcard 12: Find the derivative of $f(x) = 3x^4 + 2x^2 - 5$.

Answer: $f'(x) = 12x^3 + 4x$. Apply power rule to each term separately.

Flashcard 13: Find the derivative of $f(x) = 3x^4 + 2x^2 - 5$.

Answer: $f'(x) = 12x^3 + 4x$. Apply power rule to each term separately.

Flashcard 14: What is the derivative of $f(x) = \text{ln}(2x)$?

Answer: $f'(x) = \frac{1}{x}$. Chain rule: $\frac{d}{dx}[\ln(2x)] = \frac{1}{2x} \cdot 2 = \frac{1}{x}$.

Flashcard 15: Calculate the derivative of $f(x) = \text{cos}(x^2)$.

Answer: $f'(x) = -2x\text{sin}(x^2)$. Chain rule: $-\sin(x^2) \cdot 2x$.

Flashcard 16: Determine the derivative of $f(x) = x^4 + 3x^{-2}$.

Answer: $f'(x) = 4x^3 - 6x^{-3}$. Power rule applied to positive and negative exponents.

Flashcard 17: Identify the product rule for derivatives.

Answer: $(uv)' = u'v + uv'$. Product rule: first times derivative of second plus second times derivative of first.

Flashcard 18: Calculate the derivative of $f(x) = \text{sin}(x) + \text{cos}(x)$.

Answer: $f'(x) = \text{cos}(x) - \text{sin}(x)$. Sum rule: derivative of sum equals sum of derivatives.

Flashcard 19: What is the derivative of $\text{csc}(x)$ with respect to $x$?

Answer: $\frac{d}{dx}[\text{csc}(x)] = -\text{csc}(x)\text{cot}(x)$. Cosecant derivative involves cotangent.

Flashcard 20: State the formula for the derivative of $\text{ln}(x)$.

Answer: $\frac{d}{dx}[\text{ln}(x)] = \frac{1}{x}$. Natural log derivative is reciprocal function.

Flashcard 21: State the formula for the derivative of $\text{sin}(x)$.

Answer: $\frac{d}{dx}[\text{sin}(x)] = \text{cos}(x)$. Derivative of sine is cosine.

Flashcard 22: Calculate the derivative of $f(x) = \text{cos}^3(x)$.

Answer: $f'(x) = -3\text{cos}^2(x)\text{sin}(x)$. Chain rule: $3\cos^2(x) \cdot (-\sin(x))$.

Flashcard 23: Calculate the derivative of $f(x) = \text{cos}^3(x)$.

Answer: $f'(x) = -3\text{cos}^2(x)\text{sin}(x)$. Chain rule: $3\cos^2(x) \cdot (-\sin(x))$.

Flashcard 24: What is the derivative of $\text{cos}(x)$ with respect to $x$?

Answer: $\frac{d}{dx}[\text{cos}(x)] = -\text{sin}(x)$. Derivative of cosine is negative sine.

Flashcard 25: What is the derivative of $\text{cot}(x)$ with respect to $x$?

Answer: $\frac{d}{dx}[\text{cot}(x)] = -\text{csc}^2(x)$. Cotangent derivative is negative cosecant squared.

Flashcard 26: Identify the derivative of $f(x) = \text{csc}(x^3)$.

Answer: $f'(x) = -3x^2\text{csc}(x^3)\text{cot}(x^3)$. Chain rule with cosecant function.

Flashcard 27: Identify the derivative of $f(x) = \text{csc}(x^3)$.

Answer: $f'(x) = -3x^2\text{csc}(x^3)\text{cot}(x^3)$. Chain rule with cosecant function.

Flashcard 28: What is the derivative of $f(x) = \text{sec}(2x)$?

Answer: $f'(x) = 2\text{sec}(2x)\text{tan}(2x)$. Chain rule with secant function.

Flashcard 29: Determine the derivative of $f(x) = \frac{\text{e}^x}{x}$.

Answer: $f'(x) = \frac{\text{e}^x(x-1)}{x^2}$. Quotient rule with $u = e^x, v = x$.

Flashcard 30: What is the derivative of $f(x) = x^n$ with respect to $x$?

Answer: $f'(x) = nx^{n-1}$. Power rule: bring down exponent, reduce power by 1.

Flashcard 31: What is the derivative of $f(x) = \text{tan}(x^2)$?

Answer: $f'(x) = 2x\text{sec}^2(x^2)$. Chain rule with tangent: $\sec^2(x^2) \cdot 2x$.

Flashcard 32: What is the derivative of $e^x$ with respect to $x$?

Answer: $\frac{d}{dx}[e^x] = e^x$. Exponential function is its own derivative.

Flashcard 33: Determine the derivative of $f(x) = x^4 + 3x^{-2}$.

Answer: $f'(x) = 4x^3 - 6x^{-3}$. Power rule applied to positive and negative exponents.

Flashcard 34: Find the derivative of $f(x) = e^{2x}$ with respect to $x$.

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

Flashcard 35: State the quotient rule for derivatives.

Answer: $(u/v)' = \frac{u'v - uv'}{v^2}$. Quotient rule formula for division of functions.

Flashcard 36: Identify the chain rule for derivatives.

Answer: If $y = f(g(x))$, then $y' = f'(g(x))g'(x)$. Chain rule for composite functions.

Flashcard 37: Compute the derivative of $f(x) = x \text{ln}(x)$.

Answer: $f'(x) = \text{ln}(x) + 1$. Product rule: $1 \cdot \ln(x) + x \cdot \frac{1}{x}$.

Flashcard 38: Identify the derivative of $f(x) = x^{5/3}$.

Answer: $f'(x) = \frac{5}{3}x^{2/3}$. Power rule with fractional exponent.

Flashcard 39: State the power rule for derivatives.

Answer: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Multiply by exponent, reduce power by 1.

Flashcard 40: Determine the derivative of $f(x) = x^3 - 4x + 6$.

Answer: $f'(x) = 3x^2 - 4$. Apply power rule term by term.

Flashcard 41: Find the derivative of $f(x) = x^2 \text{sin}(x)$.

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

Flashcard 42: Determine the derivative of $f(x) = \text{sin}^2(x)$.

Answer: $f'(x) = 2\text{sin}(x)\text{cos}(x)$. Chain rule: $2\sin(x)\cos(x)$.

Flashcard 43: What is the derivative of $f(x) = \text{sec}(2x)$?

Answer: $f'(x) = 2\text{sec}(2x)\text{tan}(2x)$. Chain rule with secant function.

Flashcard 44: Compute the derivative of $f(x) = x \ln(x)$.

Answer: $f'(x) = \ln(x) + 1$. Product rule: $1 \cdot \ln(x) + x \cdot \frac{1}{x}$.

Flashcard 45: Determine the derivative of $f(x) = \frac{\text{e}^x}{x}$.

Answer: $f'(x) = \frac{\text{e}^x(x-1)}{x^2}$. Quotient rule with $u = e^x, v = x$.

Flashcard 46: Find the derivative of $f(x) = \text{sec}^2(x)$.

Answer: $f'(x) = 2\text{sec}^2(x)\text{sec}(x)\text{tan}(x)$. Chain rule: $2\sec(x) \cdot \sec(x)\tan(x)$.

Flashcard 47: What is the derivative of $f(x) = \text{ln}(x^2 + 1)$?

Answer: $f'(x) = \frac{2x}{x^2 + 1}$. Chain rule: $\frac{1}{x^2+1} \cdot 2x$.

Flashcard 48: Calculate the derivative of $f(x) = \text{tan}(3x)$.

Answer: $f'(x) = 3\text{sec}^2(3x)$. Chain rule with tangent: $\sec^2(3x) \cdot 3$.

Flashcard 49: What is the derivative of $f(x) = \text{e}^{\text{sin}(x)}$?

Answer: $f'(x) = \text{e}^{\text{sin}(x)}\text{cos}(x)$. Chain rule: $e^{\sin(x)} \cdot \cos(x)$.

Flashcard 50: What is the derivative of $\text{cot}(x)$ with respect to $x$?

Answer: $\frac{d}{dx}[\text{cot}(x)] = -\text{csc}^2(x)$. Cotangent derivative is negative cosecant squared.

Flashcard 51: Find the derivative of $f(x) = e^{2x}$ with respect to $x$.

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

Flashcard 52: State the formula for the derivative of $\text{ln}(x)$.

Answer: $\frac{d}{dx}[\text{ln}(x)] = \frac{1}{x}$. Natural log derivative is reciprocal function.

Flashcard 53: Compute the derivative of $f(x) = \frac{1}{x}$.

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

Flashcard 54: State the quotient rule for derivatives.

Answer: $(u/v)' = \frac{u'v - uv'}{v^2}$. Quotient rule formula for division of functions.

Flashcard 55: Find the derivative of $f(x) = \text{ln}(\text{sin}(x))$.

Answer: $f'(x) = \frac{\text{cos}(x)}{\text{sin}(x)}$. Chain rule: $\frac{1}{\sin(x)} \cdot \cos(x)$.

Flashcard 56: Calculate the derivative of $f(x) = 4x^5 - 2x^3 + x$.

Answer: $f'(x) = 20x^4 - 6x^2 + 1$. Power rule applied to polynomial.

Flashcard 57: Compute the derivative of $f(x) = \frac{1}{x}$.

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

Flashcard 58: Calculate the derivative of $f(x) = 4x^5 - 2x^3 + x$.

Answer: $f'(x) = 20x^4 - 6x^2 + 1$. Power rule applied to polynomial.

Flashcard 59: Identify the chain rule for derivatives.

Answer: If $y = f(g(x))$, then $y' = f'(g(x))g'(x)$. Chain rule for composite functions.

Flashcard 60: Compute the derivative of $f(x) = \frac{1}{x^2 + 1}$.

Answer: $f'(x) = -\frac{2x}{(x^2 + 1)^2}$. Quotient rule with $u = 1, v = x^2 + 1$.

Flashcard 61: Calculate the derivative of $f(x) = \sin(x) + \cos(x)$.

Answer: $f'(x) = \cos(x) - \sin(x)$. Sum rule: derivative of sum equals sum of derivatives.

Flashcard 62: State the derivative of $\text{tan}(x)$ with respect to $x$.

Answer: $\frac{d}{dx}[\text{tan}(x)] = \text{sec}^2(x)$. Derivative of tangent is secant squared.

Flashcard 63: Compute the derivative of $f(x) = \frac{1}{x^2 + 1}$.

Answer: $f'(x) = -\frac{2x}{(x^2 + 1)^2}$. Quotient rule with $u = 1, v = x^2 + 1$.

Flashcard 64: Find the derivative of $f(x) = \text{ln}(\text{sin}(x))$.

Answer: $f'(x) = \frac{\text{cos}(x)}{\text{sin}(x)}$. Chain rule: $\frac{1}{\sin(x)} \cdot \cos(x)$.

Flashcard 65: Determine the derivative of $f(x) = \text{sin}^2(x)$.

Answer: $f'(x) = 2\text{sin}(x)\text{cos}(x)$. Chain rule: $2\sin(x)\cos(x)$.

Flashcard 66: What is the derivative of $\text{csc}(x)$ with respect to $x$?

Answer: $\frac{d}{dx}[\text{csc}(x)] = -\text{csc}(x)\text{cot}(x)$. Cosecant derivative involves cotangent.

Flashcard 67: State the power rule for derivatives.

Answer: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Multiply by exponent, reduce power by 1.

Flashcard 68: Identify the derivative of $f(x) = x^{5/3}$.

Answer: $f'(x) = \frac{5}{3}x^{2/3}$. Power rule with fractional exponent.

Flashcard 69: What is the derivative of $f(x) = \text{e}^{\text{sin}(x)}$?

Answer: $f'(x) = \text{e}^{\text{sin}(x)}\text{cos}(x)$. Chain rule: $e^{\sin(x)} \cdot \cos(x)$.

Flashcard 70: State the formula for the derivative of $\text{sin}(x)$.

Answer: $\frac{d}{dx}[\text{sin}(x)] = \text{cos}(x)$. Derivative of sine is cosine.

Flashcard 71: What is the derivative of $f(x) = \text{ln}(2x)$?

Answer: $f'(x) = \frac{1}{x}$. Chain rule: $\frac{d}{dx}[\ln(2x)] = \frac{1}{2x} \cdot 2 = \frac{1}{x}$.

Flashcard 72: Find the derivative of $f(x) = x^2 \text{sin}(x)$.

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

Flashcard 73: What is the derivative of $\text{cos}(x)$ with respect to $x$?

Answer: $\frac{d}{dx}[\text{cos}(x)] = -\text{sin}(x)$. Derivative of cosine is negative sine.

Flashcard 74: What is the derivative of $f(x) = \text{ln}(x^2 + 1)$?

Answer: $f'(x) = \frac{2x}{x^2 + 1}$. Chain rule: $\frac{1}{x^2+1} \cdot 2x$.