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

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

Flashcard 2: Differentiate $f(x) = \frac{7}{x}$.

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

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

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

Flashcard 4: Find the derivative of $f(x) = \cos(3x)$.

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

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

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

Flashcard 6: What is the derivative of $e^x$?

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

Flashcard 7: 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 8: Which rule is used to differentiate $f(x) = (3x^2)(\ln(x))$?

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

Flashcard 9: Differentiate $f(x) = \cot(x)$ using trigonometric rules.

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

Flashcard 10: What is the derivative of $\ln(5x)$?

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

Flashcard 11: Differentiate $f(x) = \sqrt{x}$.

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

Flashcard 12: What is the derivative of $\tan(x)$?

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

Flashcard 13: 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 14: 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 15: Find the derivative of $f(x) = 5x^3 + 2x - 7$.

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

Flashcard 16: Differentiate $f(x) = \frac{1}{x}$ using the power rule.

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

Flashcard 17: Identify the derivative of $f(x) = \csc(x)$.

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

Flashcard 18: Differentiate $f(x) = \sec(x)$ using trigonometric differentiation.

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

Flashcard 19: 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 20: Differentiate $f(x) = 5e^{2x}$.

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

Flashcard 21: 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 22: Differentiate $f(x) = \ln(x^2 + 1)$.

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

Flashcard 23: Differentiate $f(x) = \sin(2x)$.

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

Flashcard 24: 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 25: State the product rule for derivatives.

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

Flashcard 26: 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 27: What is the derivative of $\cot(x)$?

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

Flashcard 28: 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 29: Identify the rule used for $f(x) = \frac{x^2 + 1}{x - 1}$.

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

Flashcard 30: What is the derivative of $\sec(x)$?

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