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AP Calculus BC Flashcards: Chain Rule

Study Chain Rule in AP Calculus BC with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

← Back to flashcard decks

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.

AP Calculus BC Flashcards: Chain Rule

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QUESTION

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

Tap or drag to reveal answer

ANSWER

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

Swipe Right = I Know It! 🎉

Swipe Left = Still Learning

All flashcards

Flashcard 1: Identify $u$ in $h(x) = (3x^2 + 2x)^5$ .

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

Flashcard 2: 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 3: 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 4: 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 5: Find $dy/dx$ if $y = \text{ln}(\text{e}^{x^2})$ .

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

Flashcard 6: Identify $u$ in $y = (\text{ln}(x))^4$ .

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

Flashcard 7: 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 8: 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 9: What is $y'$ if $y = \text{cos}(5x)$ ?

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

Flashcard 10: Differentiate $y = \text{e}^{3x+7}$ .

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

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

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

Flashcard 12: 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 13: 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 14: 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 15: 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 16: 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 17: 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 18: 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 19: 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 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: Differentiate $y = (2x + 5)^6$ .

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

Flashcard 22: 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 23: Differentiate $e^{x^2}$ using the Chain Rule.

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

Flashcard 24: 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 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: 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 27: 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 28: 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 29: 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 30: 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 $f(g(x))$ .

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AP Calculus BC Flashcards: Chain Rule

Study Chain Rule in AP Calculus BC with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

← Back to flashcard decks

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.

AP Calculus BC Flashcards: Chain Rule

/ 30

0 reviewed

0% Complete

0 reviewing

QUESTION

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

Tap or drag to reveal answer

ANSWER

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

Swipe Right = I Know It! 🎉

Swipe Left = Still Learning

All flashcards

Flashcard 1: Identify $u$ in $h(x) = (3x^2 + 2x)^5$ .

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

Flashcard 2: 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 3: 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 4: 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 5: Find $dy/dx$ if $y = \text{ln}(\text{e}^{x^2})$ .

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

Flashcard 6: Identify $u$ in $y = (\text{ln}(x))^4$ .

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

Flashcard 7: 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 8: 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 9: What is $y'$ if $y = \text{cos}(5x)$ ?

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

Flashcard 10: Differentiate $y = \text{e}^{3x+7}$ .

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

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

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

Flashcard 12: 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 13: 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 14: 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 15: 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 16: 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 17: 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 18: 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 19: 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 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: Differentiate $y = (2x + 5)^6$ .

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

Flashcard 22: 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 23: Differentiate $e^{x^2}$ using the Chain Rule.

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

Flashcard 24: 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 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: 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 27: 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 28: 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 29: 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 30: 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 $f(g(x))$ .

Opening subject page...

AP Calculus BC Flashcards: Chain Rule

What this deck covers

How to use these flashcards

AP Calculus BC Flashcards: Chain Rule

All flashcards

Flashcard 1: Identify uuu in h(x)=(3x2+2x)5h(x) = (3x^2 + 2x)^5h(x)=(3x2+2x)5.

Flashcard 2: Find the derivative of sin2(x)\text{sin}^2(x)sin2(x).

Flashcard 3: Differentiate y=sin(ln(x2))y = \text{sin}(\text{ln}(x^2))y=sin(ln(x2)).

Flashcard 4: Differentiate f(x)=ln(ex+1)f(x) = \text{ln}(\text{e}^x + 1)f(x)=ln(ex+1).

Flashcard 5: Find dy/dxdy/dxdy/dx if y=ln(ex2)y = \text{ln}(\text{e}^{x^2})y=ln(ex2).

Flashcard 6: Identify uuu in y=(ln(x))4y = (\text{ln}(x))^4y=(ln(x))4.

Flashcard 7: Differentiate y=etan(x)y = \text{e}^{\text{tan}(x)}y=etan(x).

Flashcard 8: Which function is the inner function in f(g(x))=e3x+7f(g(x)) = \text{e}^{3x+7}f(g(x))=e3x+7?

Flashcard 9: What is y′y'y′ if y=cos(5x)y = \text{cos}(5x)y=cos(5x)?

Flashcard 10: Differentiate y=e3x+7y = \text{e}^{3x+7}y=e3x+7.

Flashcard 11: Differentiate y=eln(x)y = \text{e}^{\text{ln}(x)}y=eln(x).

Flashcard 12: Differentiate y=cos−1(5x)y = \text{cos}^{-1}(5x)y=cos−1(5x).

Flashcard 13: Find f′(x)f'(x)f′(x) if f(x)=cos(ex)f(x) = \text{cos}(\text{e}^x)f(x)=cos(ex).

Flashcard 14: Find d/dxd/dxd/dx of y=cos(e2x)y = \text{cos}(\text{e}^{2x})y=cos(e2x).

Flashcard 15: Differentiate y=tan2(3x)y = \text{tan}^2(3x)y=tan2(3x) using the Chain Rule.

Flashcard 16: Differentiate f(x)=ex3f(x) = \text{e}^{x^3}f(x)=ex3.

Flashcard 17: Differentiate f(x)=cos3(x)f(x) = \text{cos}^3(x)f(x)=cos3(x).

Flashcard 18: Which function is the outer function in f(g(x))=e3x+7f(g(x)) = \text{e}^{3x+7}f(g(x))=e3x+7?

Flashcard 19: Evaluate ddx[ln(x2+1)]\frac{d}{dx}[\text{ln}(x^2 + 1)]dxd​[ln(x2+1)].

Flashcard 20: Identify the outer function in f(g(x))=tan⁡(x23)f(g(x)) = \tan(\frac{x^2}{3})f(g(x))=tan(3x2​).

Flashcard 21: Differentiate y=(2x+5)6y = (2x + 5)^6y=(2x+5)6.

Flashcard 22: Differentiate g(x)=ln(4x2+1)g(x) = \text{ln}(4x^2 + 1)g(x)=ln(4x2+1).

Flashcard 23: Differentiate ex2e^{x^2}ex2 using the Chain Rule.

Flashcard 24: Identify the inner function in f(g(x))=tan⁡(x23)f(g(x)) = \tan(\frac{x^2}{3})f(g(x))=tan(3x2​).

Flashcard 25: State the formula for the Chain Rule.

Flashcard 26: Find y′y'y′ if y=tan(ln(x))y = \text{tan}(\text{ln}(x))y=tan(ln(x)).

Flashcard 27: Differentiate y=(ln(x))4y = (\text{ln}(x))^4y=(ln(x))4 using the Chain Rule.

Flashcard 28: Find f′(x)f'(x)f′(x) if f(x)=sin(x3)f(x) = \text{sin}(x^3)f(x)=sin(x3).

Flashcard 29: Differentiate f(x)=esin(x)f(x) = \text{e}^{\text{sin}(x)}f(x)=esin(x).

Flashcard 30: What is the derivative of f(g(x))f(g(x))f(g(x)) using the Chain Rule?

Opening subject page...

AP Calculus BC Flashcards: Chain Rule

What this deck covers

How to use these flashcards

AP Calculus BC Flashcards: Chain Rule

All flashcards

Flashcard 1: Identify uuu in h(x)=(3x2+2x)5h(x) = (3x^2 + 2x)^5h(x)=(3x2+2x)5.

Flashcard 2: Find the derivative of sin2(x)\text{sin}^2(x)sin2(x).

Flashcard 3: Differentiate y=sin(ln(x2))y = \text{sin}(\text{ln}(x^2))y=sin(ln(x2)).

Flashcard 4: Differentiate f(x)=ln(ex+1)f(x) = \text{ln}(\text{e}^x + 1)f(x)=ln(ex+1).

Flashcard 5: Find dy/dxdy/dxdy/dx if y=ln(ex2)y = \text{ln}(\text{e}^{x^2})y=ln(ex2).

Flashcard 6: Identify uuu in y=(ln(x))4y = (\text{ln}(x))^4y=(ln(x))4.

Flashcard 7: Differentiate y=etan(x)y = \text{e}^{\text{tan}(x)}y=etan(x).

Flashcard 8: Which function is the inner function in f(g(x))=e3x+7f(g(x)) = \text{e}^{3x+7}f(g(x))=e3x+7?

Flashcard 9: What is y′y'y′ if y=cos(5x)y = \text{cos}(5x)y=cos(5x)?

Flashcard 10: Differentiate y=e3x+7y = \text{e}^{3x+7}y=e3x+7.

Flashcard 11: Differentiate y=eln(x)y = \text{e}^{\text{ln}(x)}y=eln(x).

Flashcard 12: Differentiate y=cos−1(5x)y = \text{cos}^{-1}(5x)y=cos−1(5x).

Flashcard 13: Find f′(x)f'(x)f′(x) if f(x)=cos(ex)f(x) = \text{cos}(\text{e}^x)f(x)=cos(ex).

Flashcard 14: Find d/dxd/dxd/dx of y=cos(e2x)y = \text{cos}(\text{e}^{2x})y=cos(e2x).

Flashcard 15: Differentiate y=tan2(3x)y = \text{tan}^2(3x)y=tan2(3x) using the Chain Rule.

Flashcard 16: Differentiate f(x)=ex3f(x) = \text{e}^{x^3}f(x)=ex3.

Flashcard 17: Differentiate f(x)=cos3(x)f(x) = \text{cos}^3(x)f(x)=cos3(x).

Flashcard 18: Which function is the outer function in f(g(x))=e3x+7f(g(x)) = \text{e}^{3x+7}f(g(x))=e3x+7?

Flashcard 19: Evaluate ddx[ln(x2+1)]\frac{d}{dx}[\text{ln}(x^2 + 1)]dxd​[ln(x2+1)].

Flashcard 20: Identify the outer function in f(g(x))=tan⁡(x23)f(g(x)) = \tan(\frac{x^2}{3})f(g(x))=tan(3x2​).

Flashcard 21: Differentiate y=(2x+5)6y = (2x + 5)^6y=(2x+5)6.

Flashcard 22: Differentiate g(x)=ln(4x2+1)g(x) = \text{ln}(4x^2 + 1)g(x)=ln(4x2+1).

Flashcard 23: Differentiate ex2e^{x^2}ex2 using the Chain Rule.

Flashcard 24: Identify the inner function in f(g(x))=tan⁡(x23)f(g(x)) = \tan(\frac{x^2}{3})f(g(x))=tan(3x2​).

Flashcard 25: State the formula for the Chain Rule.

Flashcard 26: Find y′y'y′ if y=tan(ln(x))y = \text{tan}(\text{ln}(x))y=tan(ln(x)).

Flashcard 27: Differentiate y=(ln(x))4y = (\text{ln}(x))^4y=(ln(x))4 using the Chain Rule.

Flashcard 28: Find f′(x)f'(x)f′(x) if f(x)=sin(x3)f(x) = \text{sin}(x^3)f(x)=sin(x3).

Flashcard 29: Differentiate f(x)=esin(x)f(x) = \text{e}^{\text{sin}(x)}f(x)=esin(x).

Flashcard 30: What is the derivative of f(g(x))f(g(x))f(g(x)) using the Chain Rule?

Flashcard 1: Identify $u$ in $h(x) = (3x^2 + 2x)^5$ .

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

Flashcard 3: Differentiate $y = \text{sin}(\text{ln}(x^2))$ .

Flashcard 4: Differentiate $f(x) = \text{ln}(\text{e}^x + 1)$ .

Flashcard 5: Find $dy/dx$ if $y = \text{ln}(\text{e}^{x^2})$ .

Flashcard 6: Identify $u$ in $y = (\text{ln}(x))^4$ .

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

Flashcard 8: Which function is the inner function in $f(g(x)) = \text{e}^{3x+7}$ ?

Flashcard 9: What is $y'$ if $y = \text{cos}(5x)$ ?

Flashcard 10: Differentiate $y = \text{e}^{3x+7}$ .

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

Flashcard 12: Differentiate $y = \text{cos}^{-1}(5x)$ .

Flashcard 13: Find $f'(x)$ if $f(x) = \text{cos}(\text{e}^x)$ .

Flashcard 14: Find $d/dx$ of $y = \text{cos}(\text{e}^{2x})$ .

Flashcard 15: Differentiate $y = \text{tan}^2(3x)$ using the Chain Rule.

Flashcard 16: Differentiate $f(x) = \text{e}^{x^3}$ .

Flashcard 17: Differentiate $f(x) = \text{cos}^3(x)$ .

Flashcard 18: Which function is the outer function in $f(g(x)) = \text{e}^{3x+7}$ ?

Flashcard 19: Evaluate $\frac{d}{dx}[\text{ln}(x^2 + 1)]$ .

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

Flashcard 21: Differentiate $y = (2x + 5)^6$ .

Flashcard 22: Differentiate $g(x) = \text{ln}(4x^2 + 1)$ .

Flashcard 23: Differentiate $e^{x^2}$ using the Chain Rule.

Flashcard 24: Identify the inner function in $f(g(x)) = \tan(\frac{x^2}{3})$ .

Flashcard 26: Find $y'$ if $y = \text{tan}(\text{ln}(x))$ .

Flashcard 27: Differentiate $y = (\text{ln}(x))^4$ using the Chain Rule.

Flashcard 28: Find $f'(x)$ if $f(x) = \text{sin}(x^3)$ .

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

Flashcard 30: What is the derivative of $f(g(x))$ using the Chain Rule?

Flashcard 1: Identify $u$ in $h(x) = (3x^2 + 2x)^5$ .

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

Flashcard 3: Differentiate $y = \text{sin}(\text{ln}(x^2))$ .

Flashcard 4: Differentiate $f(x) = \text{ln}(\text{e}^x + 1)$ .

Flashcard 5: Find $dy/dx$ if $y = \text{ln}(\text{e}^{x^2})$ .

Flashcard 6: Identify $u$ in $y = (\text{ln}(x))^4$ .

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

Flashcard 8: Which function is the inner function in $f(g(x)) = \text{e}^{3x+7}$ ?

Flashcard 9: What is $y'$ if $y = \text{cos}(5x)$ ?

Flashcard 10: Differentiate $y = \text{e}^{3x+7}$ .

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

Flashcard 12: Differentiate $y = \text{cos}^{-1}(5x)$ .

Flashcard 13: Find $f'(x)$ if $f(x) = \text{cos}(\text{e}^x)$ .

Flashcard 14: Find $d/dx$ of $y = \text{cos}(\text{e}^{2x})$ .

Flashcard 15: Differentiate $y = \text{tan}^2(3x)$ using the Chain Rule.

Flashcard 16: Differentiate $f(x) = \text{e}^{x^3}$ .

Flashcard 17: Differentiate $f(x) = \text{cos}^3(x)$ .

Flashcard 18: Which function is the outer function in $f(g(x)) = \text{e}^{3x+7}$ ?

Flashcard 19: Evaluate $\frac{d}{dx}[\text{ln}(x^2 + 1)]$ .

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

Flashcard 21: Differentiate $y = (2x + 5)^6$ .

Flashcard 22: Differentiate $g(x) = \text{ln}(4x^2 + 1)$ .

Flashcard 23: Differentiate $e^{x^2}$ using the Chain Rule.

Flashcard 24: Identify the inner function in $f(g(x)) = \tan(\frac{x^2}{3})$ .

Flashcard 26: Find $y'$ if $y = \text{tan}(\text{ln}(x))$ .

Flashcard 27: Differentiate $y = (\text{ln}(x))^4$ using the Chain Rule.

Flashcard 28: Find $f'(x)$ if $f(x) = \text{sin}(x^3)$ .

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

Flashcard 30: What is the derivative of $f(g(x))$ using the Chain Rule?