AP Calculus AB Flashcards: Chain Rule

What this deck covers

This deck focuses on Chain Rule, 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 $y = \text{cos}(\text{ln}(x^2))$.

Answer: $-\frac{2\text{sin}(\text{ln}(x^2))}{x}$. Cosine composition with natural log of $x^2$.

Flashcard 2: Differentiate $y = \sin^3(x)$.

Answer: $3\sin^2(x)\cos(x)$. Power rule: $3\sin^2(x) \times \cos(x)$.

Flashcard 3: Differentiate $y = e^{\tan(x)}$

Answer: $\sec^2(x) e^{\tan(x)}$. Exponential times derivative of tangent function.

Flashcard 4: Differentiate $y = \text{cos}(5x)$.

Answer: $-5\text{sin}(5x)$. Derivative of cosine is $-\sin$ times inner derivative.

Flashcard 5: Find the derivative of $y = \frac{1}{\text{e}^{2x}}$.

Answer: $-2\text{e}^{-2x}$. Rewrite as $\text{e}^{-2x}$ and differentiate.

Flashcard 6: Differentiate $y = \text{ln}(\text{e}^{2x})$.

Answer:

1. Simplifies to $\ln(\text{e}^{2x}) = 2x$.

Flashcard 7: Differentiate $y = \text{e}^{\text{e}^x}$.

Answer: $\text{e}^x\text{e}^{\text{e}^x}$. Double exponential: $\text{e}^x \times \text{e}^{\text{e}^x}$.

Flashcard 8: Differentiate $y = \text{tan}^3(x)$.

Answer: $3\text{tan}^2(x)\text{sec}^2(x)$. Power rule: $3\tan^2(x) \times \sec^2(x)$.

Flashcard 9: Differentiate $y = (5x + 3)^7$.

Answer: $35(5x + 3)^6$. Power rule: $7(5x + 3)^6 \times 5$.

Flashcard 10: Differentiate $y = \text{cos}(\text{sin}(x))$.

Answer: $-\text{sin}(\text{sin}(x))\text{cos}(x)$. Cosine composition: $-\sin(\sin(x)) \times \cos(x)$.

Flashcard 11: Differentiate $y = \text{sin}(\text{e}^{3x})$.

Answer: $3\text{e}^{3x}\text{cos}(\text{e}^{3x})$. Sine composition: $\cos(\text{e}^{3x}) \times 3\text{e}^{3x}$.

Flashcard 12: Differentiate $y = \text{tan}(\text{ln}(x))$.

Answer: $\frac{\text{sec}^2(\text{ln}(x))}{x}$. Tangent of natural log: $\sec^2(\ln(x)) \times \frac{1}{x}$.

Flashcard 13: Differentiate $y = \text{cos}(\text{ln}(x))$.

Answer: $-\frac{\text{sin}(\text{ln}(x))}{x}$. Chain rule: $-\sin(\ln(x)) \times \frac{1}{x}$.

Flashcard 14: Differentiate $y = \text{ln}(7x^2 + 5)$.

Answer: $\frac{14x}{7x^2 + 5}$. Derivative of $\ln(u)$ is $\frac{1}{u} \times u'$.

Flashcard 15: Differentiate $y = (\text{ln}(x))^3$.

Answer: $\frac{3(\text{ln}(x))^2}{x}$. Power rule: $3(\ln(x))^2 \times \frac{1}{x}$.

Flashcard 16: Differentiate $y = (\text{e}^x)^2$.

Answer: $2\text{e}^{2x}$. Use power rule: $(\text{e}^x)^2 = \text{e}^{2x}$.

Flashcard 17: Differentiate $y = \frac{1}{\text{sin}(x^2)}$.

Answer: $-\frac{2x\text{cos}(x^2)}{\text{sin}^2(x^2)}$. Use quotient rule or rewrite as $\csc(x^2)$.

Flashcard 18: Differentiate $y = \text{cos}(2x^2)$.

Answer: $-4x\text{sin}(2x^2)$. Cosine derivative: $-\sin(2x^2) \times 4x$.

Flashcard 19: Differentiate $y = \text{sin}(\text{e}^x)$.

Answer: $\text{e}^x\text{cos}(\text{e}^x)$. Cosine of $\text{e}^x$ times derivative of $\text{e}^x$.

Flashcard 20: Identify the inner function in $y = (4x^2 + 1)^5$.

Answer: $u = 4x^2 + 1$. The expression inside the power is the inner function.

Flashcard 21: What is the derivative of $y = (3x + 2)^4$ using the Chain Rule?

Answer: $12(3x + 2)^3$. Power rule with chain rule: $4(3x + 2)^3 \times 3$.

Flashcard 22: Differentiate $y = \text{tan}^2(x)$.

Answer: $2\text{tan}(x)\text{sec}^2(x)$. Power rule: $2\tan(x) \times \sec^2(x)$.

Flashcard 23: State the formula for the Chain Rule.

Answer: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Multiplies outer derivative by inner derivative.

Flashcard 24: Differentiate $y = \text{ln}(x^2 + 2x)$.

Answer: $\frac{2x + 2}{x^2 + 2x}$. Natural log derivative: $\frac{1}{x^2 + 2x} \times (2x + 2)$.

Flashcard 25: Identify the inner function in $y = \text{ln}(\text{cos}(x))$.

Answer: $u = \text{cos}(x)$. The cosine function is inside the natural log.

Flashcard 26: Differentiate $y = \text{e}^{\text{cos}(x)}$

Answer: $-\text{sin}(x)\text{e}^{\text{cos}(x)}$. Exponential times derivative of exponent: $-\sin(x)$

Flashcard 27: Differentiate $y = \text{sin}(\text{cos}(x))$.

Answer: $-\text{cos}(\text{cos}(x))\text{sin}(x)$. Composition: sine of cosine times derivative of cosine.

Flashcard 28: Differentiate $y = \tan(4x)$.

Answer: $4\sec^2(4x)$. Tangent derivative is $\sec^2$ times inner derivative.

Flashcard 29: Find $\frac{d}{dx}(\text{e}^{3x+1})$.

Answer: $3\text{e}^{3x+1}$. Exponential function times derivative of exponent.

Flashcard 30: Differentiate $y = \text{e}^{x^2 + 3}$.

Answer: $2x\text{e}^{x^2 + 3}$. Exponential function times derivative of exponent.