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 $y = (\ln(x))^3$.

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

Flashcard 2: 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 3: Differentiate $y = e^{\tan(x)}$

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

Flashcard 4: Differentiate $y = \sin^3(x)$.

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

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: 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 7: Differentiate $y = \text{sin}^4(x)$.

Answer: $4\text{sin}^3(x)\text{cos}(x)$. Power rule: $4\sin^3(x) \times \cos(x)$.

Flashcard 8: Differentiate $y = \text{e}^{2x^2 + 3x}$.

Answer: $(4x + 3)\text{e}^{2x^2 + 3x}$. Exponential function times derivative of exponent.

Flashcard 9: 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 10: 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 11: 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 12: 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 13: Find $\frac{d}{dx}(\text{e}^{3x+1})$.

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

Flashcard 14: Differentiate $y = \text{cos}(5x)$.

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

Flashcard 15: 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 16: 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 17: 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 18: Differentiate $y = \text{e}^{x^2 + 3}$.

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

Flashcard 19: Differentiate $y = \text{e}^{x^2 + 3}$.

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

Flashcard 20: Differentiate $y = \text{ln}(\text{e}^{2x})$.

Answer:

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

Flashcard 21: 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 22: Find $\frac{d}{dx}(\text{e}^{3x+1})$.

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

Flashcard 23: 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 24: Differentiate $y = (\text{e}^x)^2$.

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

Flashcard 25: 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 26: Differentiate $y = \frac{1}{(2x^2 + 3)^4}$.

Answer: $-\frac{16x}{(2x^2 + 3)^5}$. Rewrite as $(2x^2 + 3)^{-4}$ and use chain rule.

Flashcard 27: 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 28: Differentiate $y = (\text{e}^x)^2$.

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

Flashcard 29: Differentiate $y = \text{sin}^4(x)$.

Answer: $4\text{sin}^3(x)\text{cos}(x)$. Power rule: $4\sin^3(x) \times \cos(x)$.

Flashcard 30: Differentiate $y = (5x + 3)^7$.

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

Flashcard 31: Differentiate $y = \text{e}^{2x^2 + 3x}$.

Answer: $(4x + 3)\text{e}^{2x^2 + 3x}$. Exponential function times derivative of exponent.

Flashcard 32: 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 33: 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 34: 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 35: Differentiate $y = \frac{1}{(2x^2 + 3)^4}$.

Answer: $-\frac{16x}{(2x^2 + 3)^5}$. Rewrite as $(2x^2 + 3)^{-4}$ and use chain rule.

Flashcard 36: 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 37: Differentiate $y = \text{cos}(5x)$.

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

Flashcard 38: 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 39: 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 40: Find the derivative of $y = \frac{1}{\text{e}^{2x}}$.

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

Flashcard 41: 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 42: Differentiate $y = \text{e}^{\text{tan}(x)}$.

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

Flashcard 43: Differentiate $y = \text{sin}^3(x)$.

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

Flashcard 44: Differentiate $y = (5x + 3)^7$.

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

Flashcard 45: 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 46: 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 47: Differentiate $y = \text{e}^{\text{cos}(x)}$

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

Flashcard 48: 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 49: Differentiate $y = \tan(4x)$.

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

Flashcard 50: 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 51: 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 52: Differentiate $y = \text{ln}(\text{e}^{2x})$.

Answer:

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

Flashcard 53: Differentiate $y = \text{e}^{\text{cos}(x)}$.

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

Flashcard 54: 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 55: 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 56: Differentiate $y = \text{cos}(2x^2)$.

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

Flashcard 57: Differentiate $y = \text{ln}(7x^2 + 5)$.

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

Flashcard 58: 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 59: Differentiate $y = \text{cos}(2x^2)$.

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

Flashcard 60: 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 61: 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 62: 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 63: Differentiate $y = \tan^2(x)$.

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

Flashcard 64: Differentiate $y = \text{ln}(7x^2 + 5)$.

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

Flashcard 65: Differentiate $y = \tan(4x)$.

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

Flashcard 66: Differentiate $y = \text{sin}(\text{cos}(x))$.

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