AP Calculus AB 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 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: Select the rule for $f(x) = e^{3x}$.

Answer: Chain rule. Composite function requires chain rule for differentiation.

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

Answer: $f'(x) = -\csc(x)\cot(x)$. Standard trigonometric derivative formula.

Flashcard 3: Differentiate $f(x) = \ln(4x)$.

Answer: $f'(x) = \frac{1}{x}$. Constant multiple in logarithm doesn't affect derivative.

Flashcard 4: Differentiate $f(x) = (2x+1)^3$ using the chain rule.

Answer: $f'(x) = 3(2x+1)^2 \cdot 2$. Apply chain rule: outer derivative times inner derivative.

Flashcard 5: Differentiate $f(x) = (2x+1)^3$ using the chain rule.

Answer: $f'(x) = 3(2x+1)^2 \cdot 2$. Apply chain rule: outer derivative times inner derivative.

Flashcard 6: Differentiate $f(x) = \sec(x)$ using trigonometric differentiation.

Answer: $f'(x) = \sec(x)\tan(x)$. Standard trigonometric derivative formula.

Flashcard 7: Differentiate $f(x) = \cot(x)$ using trigonometric rules.

Answer: $f'(x) = -\csc^2(x)$. Standard trigonometric derivative formula.

Flashcard 8: Differentiate $f(x) = \frac{1}{x}$ using the power rule.

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

Flashcard 9: What is the derivative of $\ln(5x)$?

Answer: $f'(x) = \frac{1}{x}$. Constant multiple in logarithm doesn't affect derivative.

Flashcard 10: What is the derivative of $\tan(x)$?

Answer: $f'(x) = \sec^2(x)$. Tangent differentiates to secant squared.

Flashcard 11: Differentiate $f(x) = 5e^{2x}$.

Answer: $f'(x) = 10e^{2x}$. Chain rule: $5 \times e^{2x} \times 2$.

Flashcard 12: Find the derivative of $f(x) = 4x^4$ using the power rule.

Answer: $f'(x) = 16x^3$. Power rule: $4 \times 4 = 16$, exponent becomes $3$.

Flashcard 13: Find the derivative of $f(x) = 5x^3 + 2x - 7$.

Answer: $f'(x) = 15x^2 + 2$. Apply power rule to each term separately.

Flashcard 14: What is the derivative of $\cos(x)$?

Answer: $f'(x) = -\sin(x)$. Cosine differentiates to negative sine.

Flashcard 15: What is the derivative of $\ln(x)$?

Answer: $f'(x) = \frac{1}{x}$. Natural logarithm derivative is reciprocal function.

Flashcard 16: Identify the rule used for $f(x) = \frac{x^2 + 1}{x - 1}$.

Answer: Quotient rule. Fraction form indicates quotient rule is needed.

Flashcard 17: Find the derivative of $f(x) = \frac{1}{x^3}$.

Answer: $f'(x) = -\frac{3}{x^4}$. Rewrite as $x^{-3}$ then apply power rule.

Flashcard 18: State the product rule for derivatives.

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

Flashcard 19: What is $f'(x)$ for $f(x) = 7$ (constant function)?

Answer: $f'(x) = 0$. Derivative of any constant is zero.

Flashcard 20: What is the derivative of $e^x$?

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

Flashcard 21: What is the derivative of $f(x) = x^n$ using the power rule?

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

Flashcard 22: Find the derivative of $f(x) = \cos(3x)$.

Answer: $f'(x) = -3\sin(3x)$. Chain rule: derivative of cosine times derivative of $3x$.

Flashcard 23: Differentiate $f(x) = x^{3/2}$ using the power rule.

Answer: $f'(x) = \frac{3}{2}x^{1/2}$. Power rule: $\frac{3}{2}$ coefficient, exponent becomes $\frac{1}{2}$.

Flashcard 24: Find the derivative of $f(x) = \cos(3x)$.

Answer: $f'(x) = -3\sin(3x)$. Chain rule: derivative of cosine times derivative of $3x$.

Flashcard 25: What is the derivative of $f(x) = \frac{7}{x^2}$?

Answer: $f'(x) = -\frac{14}{x^3}$. Rewrite as $7x^{-2}$ then apply power rule.

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

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

Flashcard 27: Differentiate $f(x) = \cot(x)$ using trigonometric rules.

Answer: $f'(x) = -\csc^2(x)$. Standard trigonometric derivative formula.

Flashcard 28: Differentiate $f(x) = \ln(x^2 + 1)$.

Answer: $f'(x) = \frac{2x}{x^2 + 1}$. Chain rule with natural log and composite function.

Flashcard 29: Differentiate $f(x) = \ln(4x)$.

Answer: $f'(x) = \frac{1}{x}$. Constant multiple in logarithm doesn't affect derivative.

Flashcard 30: Find the derivative of $f(x) = \frac{1}{x^3}$.

Answer: $f'(x) = -\frac{3}{x^4}$. Rewrite as $x^{-3}$ then apply power rule.

Flashcard 31: What is the derivative of $f(x) = 3x^{-2}$?

Answer: $f'(x) = -6x^{-3}$. Power rule: $3 \times (-2) = -6$, exponent becomes $-3$.

Flashcard 32: What is the derivative of $\tan(x)$?

Answer: $f'(x) = \sec^2(x)$. Tangent differentiates to secant squared.

Flashcard 33: Differentiate $f(x) = \sqrt{x}$.

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

Flashcard 34: Which rule is used to differentiate $f(x) = (3x^2)(\ln(x))$?

Answer: Product rule. Two functions multiplied together requires product rule.

Flashcard 35: What is the derivative of $\csc(x)$?

Answer: $f'(x) = -\csc(x)\cot(x)$. Cosecant differentiates to negative cosecant cotangent.

Flashcard 36: What is the derivative of $\csc(x)$?

Answer: $f'(x) = -\csc(x)\cot(x)$. Cosecant differentiates to negative cosecant cotangent.

Flashcard 37: Differentiate $f(x) = \frac{7}{x}$.

Answer: $f'(x) = -\frac{7}{x^2}$. Rewrite as $7x^{-1}$ then apply power rule.

Flashcard 38: State the quotient rule for derivatives.

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

Flashcard 39: State the quotient rule for derivatives.

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

Flashcard 40: What is the derivative of $\cos(x)$?

Answer: $f'(x) = -\sin(x)$. Cosine differentiates to negative sine.

Flashcard 41: State the product rule for derivatives.

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

Flashcard 42: What is the derivative of $\ln(5x)$?

Answer: $f'(x) = \frac{1}{x}$. Constant multiple in logarithm doesn't affect derivative.

Flashcard 43: Find the derivative of $f(x) = 4x^4$ using the power rule.

Answer: $f'(x) = 16x^3$. Power rule: $4 \times 4 = 16$, exponent becomes $3$.

Flashcard 44: Which rule is used to differentiate $f(x) = (3x^2)(\ln(x))$?

Answer: Product rule. Two functions multiplied together requires product rule.

Flashcard 45: What is the derivative of $\sec(x)$?

Answer: $f'(x) = \sec(x)\tan(x)$. Secant differentiates to secant tangent.

Flashcard 46: What is $f'(x)$ for $f(x) = 7$ (constant function)?

Answer: $f'(x) = 0$. Derivative of any constant is zero.

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

Answer: $f'(x) = -\csc^2(x)$. Cotangent differentiates to negative cosecant squared.

Flashcard 48: Which differentiation rule applies to $f(x) = x^2 \sin(x)$?

Answer: Product rule. Polynomial times trigonometric function needs product rule.

Flashcard 49: Find the derivative of $f(x) = \frac{x^3}{3}$.

Answer: $f'(x) = x^2$. Derivative of $\frac{x^3}{3}$ using power rule on numerator.

Flashcard 50: Identify the rule used for $f(x) = \frac{x^2 + 1}{x - 1}$.

Answer: Quotient rule. Fraction form indicates quotient rule is needed.

Flashcard 51: Differentiate $f(x) = \ln(x^2 + 1)$.

Answer: $f'(x) = \frac{2x}{x^2 + 1}$. Chain rule with natural log and composite function.

Flashcard 52: What is the derivative of $\sin(x)$?

Answer: $f'(x) = \cos(x)$. Sine differentiates to cosine.

Flashcard 53: What is the derivative of $e^x$?

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

Flashcard 54: Which differentiation rule applies to $f(x) = x^2 \sin(x)$?

Answer: Product rule. Polynomial times trigonometric function needs product rule.

Flashcard 55: Differentiate $f(x) = \sin(2x)$.

Answer: $f'(x) = 2\cos(2x)$. Chain rule: derivative of sine times derivative of $2x$.

Flashcard 56: Differentiate $f(x) = \frac{1}{x}$ using the power rule.

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

Flashcard 57: What is the derivative of $f(x) = 3x^{-2}$?

Answer: $f'(x) = -6x^{-3}$. Power rule: $3 \times (-2) = -6$, exponent becomes $-3$.

Flashcard 58: What is the derivative of $f(x) = \frac{7}{x^2}$?

Answer: $f'(x) = -\frac{14}{x^3}$. Rewrite as $7x^{-2}$ then apply power rule.

Flashcard 59: Differentiate $f(x) = x^{3/2}$ using the power rule.

Answer: $f'(x) = \frac{3}{2}x^{1/2}$. Power rule: $\frac{3}{2}$ coefficient, exponent becomes $\frac{1}{2}$.

Flashcard 60: Identify the derivative of $f(x) = \csc(x)$.

Answer: $f'(x) = -\csc(x)\cot(x)$. Standard trigonometric derivative formula.

Flashcard 61: What is the derivative of $\ln(x)$?

Answer: $f'(x) = \frac{1}{x}$. Natural logarithm derivative is reciprocal function.

Flashcard 62: What is the derivative of $f(x) = x^{-3}$?

Answer: $f'(x) = -3x^{-4}$. Power rule with negative exponent: $-3 \times (-1) = 3$, new power $-4$.

Flashcard 63: What is the derivative of $\sec(x)$?

Answer: $f'(x) = \sec(x)\tan(x)$. Secant differentiates to secant tangent.

Flashcard 64: What is the derivative of $\sin(x)$?

Answer: $f'(x) = \cos(x)$. Sine differentiates to cosine.

Flashcard 65: Differentiate $f(x) = \frac{7}{x}$.

Answer: $f'(x) = -\frac{7}{x^2}$. Rewrite as $7x^{-1}$ then apply power rule.

Flashcard 66: Differentiate $f(x) = \sin(2x)$.

Answer: $f'(x) = 2\cos(2x)$. Chain rule: derivative of sine times derivative of $2x$.

Flashcard 67: What is the derivative of $f(x) = x^{-3}$?

Answer: $f'(x) = -3x^{-4}$. Power rule with negative exponent: $-3 \times (-1) = 3$, new power $-4$.

Flashcard 68: Differentiate $f(x) = \sec(x)$ using trigonometric differentiation.

Answer: $f'(x) = \sec(x)\tan(x)$. Standard trigonometric derivative formula.

Flashcard 69: Select the rule for $f(x) = e^{3x}$.

Answer: Chain rule. Composite function requires chain rule for differentiation.

Flashcard 70: Differentiate $f(x) = 5e^{2x}$.

Answer: $f'(x) = 10e^{2x}$. Chain rule: $5 \times e^{2x} \times 2$.

Flashcard 71: Find the derivative of $f(x) = \frac{x^3}{3}$.

Answer: $f'(x) = x^2$. Derivative of $\frac{x^3}{3}$ using power rule on numerator.

Flashcard 72: What is the derivative of $\cot(x)$?

Answer: $f'(x) = -\csc^2(x)$. Cotangent differentiates to negative cosecant squared.

Flashcard 73: State the chain rule for derivatives.

Answer: $(f(g(x)))' = f'(g(x))g'(x)$. Multiply derivative of outer by derivative of inner function.

Flashcard 74: What is the derivative of $f(x) = x^n$ using the power rule?

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

Flashcard 75: Find the derivative of $f(x) = 5x^3 + 2x - 7$.

Answer: $f'(x) = 15x^2 + 2$. Apply power rule to each term separately.

Flashcard 76: State the chain rule for derivatives.

Answer: $(f(g(x)))' = f'(g(x))g'(x)$. Multiply derivative of outer by derivative of inner function.