AP Calculus BC 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 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: Differentiate $y = e^{ an(x)}$.

Answer: $\sec^2(x) e^{ an(x)}$. Chain rule: $e^u \times u'$ where $u = \tan(x)$

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

Answer: $\frac{e^x}{e^x + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = e^x + 1$.

Flashcard 3: Differentiate $y = (2x + 5)^6$.

Answer: $12(2x+5)^5$. Power rule with chain rule: $6u^5 \times u'$.

Flashcard 4: Identify $u$ in $y = (\text{ln}(x))^4$.

Answer: $u = \text{ln}(x)$. The natural logarithm function raised to power 4.

Flashcard 5: Differentiate $g(x) = \text{ln}(4x^2 + 1)$.

Answer: $\frac{8x}{4x^2 + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = 4x^2 + 1$.

Flashcard 6: Evaluate $\frac{d}{dx}[\text{ln}(x^2 + 1)]$.

Answer: $\frac{2x}{x^2 + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = x^2 + 1$.

Flashcard 7: Identify the inner function in $f(g(x)) = \tan(\frac{x^2}{3})$.

Answer: $g(x) = \frac{x^2}{3}$. The expression inside the tangent function.

Flashcard 8: Differentiate $y = (\text{ln}(x))^4$ using the Chain Rule.

Answer: $\frac{4(\text{ln}(x))^3}{x}$. Power rule with chain rule: $4u^3 \times u'$.

Flashcard 9: Differentiate $y = \text{sin}(\text{ln}(x^2))$.

Answer: $\frac{2x\text{cos}(\text{ln}(x^2))}{x^2}$. Chain rule twice: $\cos(\ln(x^2)) \times \frac{2x}{x^2}$.

Flashcard 10: Find $d/dx$ of $y = \text{cos}(\text{e}^{2x})$.

Answer: $-2\text{e}^{2x}\text{sin}(\text{e}^{2x})$. Chain rule: $-\sin(u) \times u'$ where $u = e^{2x}$.

Flashcard 11: Differentiate $g(x) = \text{ln}(4x^2 + 1)$.

Answer: $\frac{8x}{4x^2 + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = 4x^2 + 1$.

Flashcard 12: Find $f'(x)$ if $f(x) = \text{sin}(x^3)$.

Answer: $3x^2 \text{cos}(x^3)$. Chain rule: $\cos(u) \times u'$ where $u = x^3$.

Flashcard 13: Find $y'$ if $y = \text{tan}(\text{ln}(x))$.

Answer: $\frac{1}{x\text{cos}^2(\text{ln}(x))}$. Chain rule twice: $\sec^2(\ln(x)) \times \frac{1}{x}$.

Flashcard 14: Find $f'(x)$ if $f(x) = \text{sin}(x^3)$.

Answer: $3x^2 \text{cos}(x^3)$. Chain rule: $\cos(u) \times u'$ where $u = x^3$.

Flashcard 15: What is $y'$ if $y = \text{cos}(5x)$?

Answer: $-5\text{sin}(5x)$. Chain rule: $-\sin(u) \times u'$ where $u = 5x$.

Flashcard 16: Identify the inner function in $f(g(x)) = \tan(\frac{x^2}{3})$.

Answer: $g(x) = \frac{x^2}{3}$. The expression inside the tangent function.

Flashcard 17: Find the derivative of $\text{sin}^2(x)$.

Answer: $2\text{sin}(x)\text{cos}(x)$. Chain rule: $2\sin(x) \times \cos(x)$ or $\sin(2x)$.

Flashcard 18: Find $d/dx$ of $y = \text{cos}(\text{e}^{2x})$.

Answer: $-2\text{e}^{2x}\text{sin}(\text{e}^{2x})$. Chain rule: $-\sin(u) \times u'$ where $u = e^{2x}$.

Flashcard 19: Identify $u$ in $h(x) = (3x^2 + 2x)^5$.

Answer: $u = 3x^2 + 2x$. The expression being raised to the 5th power.

Flashcard 20: Identify the outer function in $f(g(x)) = \tan(\frac{x^2}{3})$.

Answer: $f(u) = \tan(u)$. The tangent function wraps the inner expression.

Flashcard 21: Find $f'(x)$ if $f(x) = \text{cos}(\text{e}^x)$.

Answer: $-\text{e}^x\text{sin}(\text{e}^x)$. Chain rule: $-\sin(u) \times u'$ where $u = e^x$.

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

Answer: $\frac{1}{x}$. Simplifies to $x$ since $e^{\ln(x)} = x$.

Flashcard 23: Find $g'(x)$ if $g(x) = \text{tan}^{-1}(2x^3)$.

Answer: $\frac{6x^2}{1 + 4x^6}$. Chain rule: $\frac{1}{1+u^2} \times u'$ where $u = 2x^3$.

Flashcard 24: Differentiate $e^{x^2}$ using the Chain Rule.

Answer: $2xe^{x^2}$. Chain rule: derivative of $e^u$ is $e^u$ times $u'$.

Flashcard 25: State the formula for the Chain Rule.

Answer: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Fundamental chain rule formula for composite functions.

Flashcard 26: Differentiate $y = \text{e}^{3x+7}$.

Answer: $3\text{e}^{3x+7}$. Chain rule: $e^u \times u'$ where $u = 3x + 7$.

Flashcard 27: Find $dy/dx$ if $y = \text{ln}(\text{e}^{x^2})$.

Answer: $2x$. Simplifies to $x^2$ since $\ln$ and $e$ cancel.

Flashcard 28: Differentiate $f(x) = \text{e}^{\text{sin}(x)}$.

Answer: $\text{cos}(x)\text{e}^{\text{sin}(x)}$. Chain rule: $e^u \times u'$ where $u = \sin(x)$.

Flashcard 29: Identify $u$ in $y = \text{tan}^2(3x)$.

Answer: $u = \text{tan}(3x)$. The squared tangent of $3x$.

Flashcard 30: Differentiate $f(x) = \text{cos}^3(x)$.

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

Flashcard 31: What is the derivative of $f(g(x))$ using the Chain Rule?

Answer: $f'(g(x)) \times g'(x)$. Chain rule applied to general composite function $f(g(x))$.

Flashcard 32: Find $g'(x)$ if $g(x) = \text{tan}^{-1}(2x^3)$.

Answer: $\frac{6x^2}{1 + 4x^6}$. Chain rule: $\frac{1}{1+u^2} \times u'$ where $u = 2x^3$.

Flashcard 33: Find $y'$ if $y = \text{tan}(\text{ln}(x))$.

Answer: $\frac{1}{x\text{cos}^2(\text{ln}(x))}$. Chain rule twice: $\sec^2(\ln(x)) \times \frac{1}{x}$.

Flashcard 34: Differentiate $y = \text{tan}^2(3x)$ using the Chain Rule.

Answer: $6\text{tan}(3x)\text{sec}^2(3x)$. Chain rule: $2u \times u'$ where $u = \tan(3x)$.

Flashcard 35: Differentiate $f(x) = \text{e}^{\text{sin}(x)}$.

Answer: $\text{cos}(x)\text{e}^{\text{sin}(x)}$. Chain rule: $e^u \times u'$ where $u = \sin(x)$.

Flashcard 36: Differentiate $y = (\text{ln}(x))^4$ using the Chain Rule.

Answer: $\frac{4(\text{ln}(x))^3}{x}$. Power rule with chain rule: $4u^3 \times u'$.

Flashcard 37: Find $f'(x)$ if $f(x) = \text{cos}(\text{e}^x)$.

Answer: $-\text{e}^x\text{sin}(\text{e}^x)$. Chain rule: $-\sin(u) \times u'$ where $u = e^x$.

Flashcard 38: Differentiate $y = (2x + 5)^6$.

Answer: $12(2x+5)^5$. Power rule with chain rule: $6u^5 \times u'$.

Flashcard 39: Differentiate $y = \text{cos}^{-1}(5x)$.

Answer: $-\frac{5}{\text{sqrt}(1 - 25x^2)}$. Chain rule: $-\frac{1}{\sqrt{1-u^2}} \times u'$ where $u = 5x$.

Flashcard 40: Identify $u$ in $y = (\text{ln}(x))^4$.

Answer: $u = \text{ln}(x)$. The natural logarithm function raised to power 4.

Flashcard 41: Which function is the outer function in $f(g(x)) = \text{e}^{3x+7}$?

Answer: $f(u) = \text{e}^u$. The exponential function wraps the linear expression.

Flashcard 42: Differentiate $f(x) = \text{e}^{x^3}$.

Answer: $3x^2\text{e}^{x^3}$. Chain rule: $e^u \times u'$ where $u = x^3$.

Flashcard 43: What is $y'$ if $y = \text{cos}(5x)$?

Answer: $-5\text{sin}(5x)$. Chain rule: $-\sin(u) \times u'$ where $u = 5x$.

Flashcard 44: Differentiate $y = \text{cos}^{-1}(5x)$.

Answer: $-\frac{5}{\text{sqrt}(1 - 25x^2)}$. Chain rule: $-\frac{1}{\sqrt{1-u^2}} \times u'$ where $u = 5x$.

Flashcard 45: Evaluate $\frac{d}{dx}[\text{ln}(x^2 + 1)]$.

Answer: $\frac{2x}{x^2 + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = x^2 + 1$.

Flashcard 46: Differentiate $y = \text{e}^{3x+7}$.

Answer: $3\text{e}^{3x+7}$. Chain rule: $e^u \times u'$ where $u = 3x + 7$.

Flashcard 47: What is the derivative of $f(g(x))$ using the Chain Rule?

Answer: $f'(g(x)) \times g'(x)$. Chain rule applied to general composite function $f(g(x))$.

Flashcard 48: Differentiate $f(x) = \text{ln}(\text{e}^x + 1)$.

Answer: $\frac{\text{e}^x}{\text{e}^x + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = e^x + 1$.

Flashcard 49: Differentiate $f(x) = \text{cos}^3(x)$.

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

Flashcard 50: Differentiate $y = \text{e}^{\text{tan}(x)}$.

Answer: $\text{sec}^2(x)\text{e}^{\text{tan}(x)}$. Chain rule: $e^u \times u'$ where $u = \tan(x)$.

Flashcard 51: Differentiate $y = \tan^2(3x)$ using the Chain Rule.

Answer: $6\tan(3x)\sec^2(3x)$. Chain rule: $2u \times u'$ where $u = \tan(3x)$.

Flashcard 52: State the formula for the Chain Rule.

Answer: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Fundamental chain rule formula for composite functions.

Flashcard 53: Which function is the inner function in $f(g(x)) = \text{e}^{3x+7}$?

Answer: $g(x) = 3x+7$. The linear expression in the exponent.

Flashcard 54: Identify $u$ in $h(x) = (3x^2 + 2x)^5$.

Answer: $u = 3x^2 + 2x$. The expression being raised to the 5th power.

Flashcard 55: Differentiate $e^{x^2}$ using the Chain Rule.

Answer: $2xe^{x^2}$. Chain rule: derivative of $e^u$ is $e^u$ times $u'$ .

Flashcard 56: Find $dy/dx$ if $y = \text{ln}(\text{e}^{x^2})$.

Answer: $2x$. Simplifies to $x^2$ since $\ln$ and $e$ cancel.

Flashcard 57: Differentiate $y = \text{e}^{\text{ln}(x)}$.

Answer: $\frac{1}{x}$. Simplifies to $x$ since $e^{\ln(x)} = x$.

Flashcard 58: Identify the outer function in $f(g(x)) = \tan(\frac{x^2}{3})$.

Answer: $f(u) = \tan(u)$. The tangent function wraps the inner expression.

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

Answer: $\frac{2x\text{cos}(\text{ln}(x^2))}{x^2}$. Chain rule twice: $\cos(\ln(x^2)) \times \frac{2x}{x^2}$.

Flashcard 60: Find the derivative of $\text{sin}^2(x)$.

Answer: $2\text{sin}(x)\text{cos}(x)$. Chain rule: $2\sin(x) \times \cos(x)$ or $\sin(2x)$.

Flashcard 61: Identify $u$ in $y = \text{tan}^2(3x)$.

Answer: $u = \text{tan}(3x)$. The squared tangent of $3x$.

Flashcard 62: Differentiate $f(x) = \text{e}^{x^3}$.

Answer: $3x^2\text{e}^{x^3}$. Chain rule: $e^u \times u'$ where $u = x^3$.

Flashcard 63: Which function is the outer function in $f(g(x)) = \text{e}^{3x+7}$?

Answer: $f(u) = \text{e}^u$. The exponential function wraps the linear expression.

Flashcard 64: Which function is the inner function in $f(g(x)) = \text{e}^{3x+7}$?

Answer: $g(x) = 3x+7$. The linear expression in the exponent.