AP Precalculus Flashcards: Rates Of Change

Study Rates Of Change in AP Precalculus with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Precalculus

Rates Of Change

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QUESTION

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If $y = 3x^3 - 5x$ , what is $\frac{dy}{dx}$ ?

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ANSWER

$\frac{dy}{dx} = 9x^2 - 5$ . Use power rule: $3 \cdot 3x^2 - 5 \cdot 1$ .

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What this deck covers

This deck focuses on Rates Of Change, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

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.

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Flashcard 1: If $y = 3x^3 - 5x$ , what is $\frac{dy}{dx}$ ?

Answer: $\frac{dy}{dx} = 9x^2 - 5$ . Use power rule: $3 \cdot 3x^2 - 5 \cdot 1$ .

Flashcard 2: What is the derivative of $f(x) = \log_a(x)$ ?

Answer: $f'(x) = \frac{1}{x \ln(a)}$ . Change of base gives factor $\frac{1}{\ln(a)}$ .

Flashcard 3: Find the derivative of $f(x) = \text{sec}(x)$ .

Answer: $f'(x) = \text{sec}(x)\text{tan}(x)$ . Product rule applied to $\sec(x) = \frac{1}{\cos(x)}$ .

Flashcard 4: What is the derivative of $f(x) = \text{cos}^2(x)$ ?

Answer: $f'(x) = -2\text{sin}(x)\text{cos}(x)$ . Chain rule gives negative of $\sin^2(x)$ derivative.

Flashcard 5: State the derivative of $f(x) = \text{arcsin}(x)$ .

Answer: $f'(x) = \frac{1}{\sqrt{1-x^2)}$ . Standard inverse trig derivative formula.

Flashcard 6: Identify the derivative of $f(x) = \text{tan}^2(x)$ .

Answer: $f'(x) = 2\text{tan}(x)\text{sec}^2(x)$ . Chain rule applied to $\tan^2(x)$ .

Flashcard 7: Identify the derivative of $f(x) = e^x$ .

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

Flashcard 8: What is the derivative of $f(x) = \text{arccsc}(x)$ ?

Answer: $f'(x) = -\frac{1}{|x|\text{sqrt}(x^2-1)}$ . Negative of arcsecant derivative.

Flashcard 9: State the derivative of $f(x) = \text{sin}^2(x)$ .

Answer: $f'(x) = 2\text{sin}(x)\text{cos}(x)$ . Use chain rule on $\sin^2(x)$ .

Flashcard 10: What is the rate of change of $f(x) = 7$ ?

Answer: Rate of change is 0. Constant functions have zero rate of change.

Flashcard 11: What is the rate of change of $f(x) = x^4$ at $x = 2$ ?

Answer: Rate of change is 32. Use $f'(x) = 4x^3$ , evaluate at $x = 2$ .

Flashcard 12: What is the rate of change of $f(x) = x^2 + 3x$ at $x=1$ ?

Answer: Rate of change is 5. Use $f'(x) = 2x + 3$ , evaluate at $x = 1$ .

Flashcard 13: What is the derivative of $f(x) = \frac{1}{x^3}$ ?

Answer: $f'(x) = -\frac{3}{x^4}$ . Rewrite as $x^{-3}$ and apply power rule.

Flashcard 14: What is the derivative of $f(x) = \text{ln}(x^2)$ ?

Answer: $f'(x) = \frac{2}{x}$ . Use chain rule: $\frac{1}{x^2} \cdot 2x = \frac{2}{x}$ .

Flashcard 15: Identify the derivative of $f(x) = \text{tan}(x)$ .

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

Flashcard 16: Calculate $\frac{d}{dx}[5x^2]$ .

Answer: $10x$ . Factor out constant 5, apply power rule.

Flashcard 17: State the constant rule for derivatives.

Answer: Derivative of a constant is 0. Constants have zero slope everywhere.

Flashcard 18: What is the derivative of $f(x) = \text{log}_a(x)$ ?

Answer: $f'(x) = \frac{1}{x\text{ln}(a)}$ . Change of base gives factor $\frac{1}{\ln(a)}$ .

Flashcard 19: Determine the derivative of $f(x) = 5^x$ .

Answer: $f'(x) = 5^x\text{ln}(5)$ . General formula $a^x$ derivative includes $\ln(a)$ factor.

Flashcard 20: Find $f'(x)$ for $f(x) = \text{sin}(x)$ .

Answer: $f'(x) = \text{cos}(x)$ . Sine differentiates to cosine.

Flashcard 21: What is the derivative of $f(x) = \text{arccos}(x)$ ?

Answer: $f'(x) = -\frac{1}{\text{sqrt}(1-x^2)}$ . Negative of arcsine derivative.

Flashcard 22: What is the derivative of $f(x) = x^x$ ?

Answer: $f'(x) = x^x(\text{ln}(x)+1)$ . Use logarithmic differentiation for variable base and exponent.

Flashcard 23: State the constant rule for derivatives.

Answer: Derivative of a constant is 0. Constants have zero slope everywhere.

Flashcard 24: What is the rate of change of $f(x) = x^2 + 3x$ at $x=1$ ?

Answer: Rate of change is 5. Use $f'(x) = 2x + 3$ , evaluate at $x = 1$ .

Flashcard 25: Determine the derivative of $f(x) = 5^x$ .

Answer: $f'(x) = 5^x \ln(5)$ . General formula $a^x$ derivative includes $\ln(a)$ factor.

Flashcard 26: What is the derivative of $f(x) = \text{cos}(x)$ ?

Answer: $f'(x) = -\text{sin}(x)$ . Cosine differentiates to negative sine.

Flashcard 27: What is the derivative of $f(x) = \text{arccos}(x)$ ?

Answer: $f'(x) = -\frac{1}{\text{sqrt}(1-x^2)}$ . Negative of arcsine derivative.

Flashcard 28: Calculate $\frac{d}{dx}[x^5 - 2x^3]$ .

Answer: $5x^4 - 6x^2$ . Apply power rule to each term separately.

Flashcard 29: If $y = 3x^3 - 5x$ , what is $\frac{dy}{dx}$ ?

Answer: $\frac{dy}{dx} = 9x^2 - 5$ . Use power rule: $3 \cdot 3x^2 - 5 \cdot 1$ .

Flashcard 30: Calculate $\frac{d}{dx}[5x^2]$ .

Answer: $10x$ . Factor out constant 5, apply power rule.

Flashcard 31: Define average rate of change of a function $f(x)$ over $[a, b]$ .

Answer: $\frac{f(b) - f(a)}{b - a}$ . Slope formula between two points on the function.

Flashcard 32: What is the derivative of $f(x) = \text{csc}(x)$ ?

Answer: $f'(x) = -\text{csc}(x)\text{cot}(x)$ . Quotient rule applied to $\csc(x) = \frac{1}{\sin(x)}$ .

Flashcard 33: What is the rate of change of $f(x) = x^4$ at $x = 2$ ?

Answer: Rate of change is 32. Use $f'(x) = 4x^3$ , evaluate at $x = 2$ .

Flashcard 34: What is the instantaneous rate of change at $x = a$ ?

Answer: The derivative $f'(a)$ . Limit of average rate of change as interval approaches zero.

Flashcard 35: State the derivative of $f(x) = \text{sin}^2(x)$ .

Answer: $f'(x) = 2\text{sin}(x)\text{cos}(x)$ . Use chain rule on $\sin^2(x)$ .

Flashcard 36: What is the derivative of $f(x) = x^x$ ?

Answer: $f'(x) = x^x(\text{ln}(x)+1)$ . Use logarithmic differentiation for variable base and exponent.

Flashcard 37: Determine the derivative of $f(x) = \text{cot}(x)$ .

Answer: $f'(x) = -\text{csc}^2(x)$ . Cotangent is $\frac{\cos(x)}{\sin(x)}$ , use quotient rule.

Flashcard 38: What is the derivative of $f(x) = x^2$ ?

Answer: $f'(x) = 2x$ . Apply power rule: bring down exponent 2, reduce power by 1.

Flashcard 39: What is the derivative of $f(x) = \text{ln}(x^2)$ ?

Answer: $f'(x) = \frac{2}{x}$ . Use chain rule: $\frac{1}{x^2} \cdot 2x = \frac{2}{x}$ .

Flashcard 40: What is the derivative of $f(x) = \frac{1}{x}$ ?

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

Flashcard 41: State the derivative of $f(x) = \text{arcsin}(x)$ .

Answer: $f'(x) = \frac{1}{\text{sqrt}(1-x^2)}$ . Standard inverse trig derivative formula.

Flashcard 42: State the power rule for derivatives.

Answer: $\frac{d}{dx}[x^n] = nx^{n-1}$ . Multiply by exponent, then decrease exponent by 1.

Flashcard 43: Calculate the derivative of $f(x) = \text{e}^{2x}$ .

Answer: $f'(x) = 2\text{e}^{2x}$ . Chain rule: multiply by inner derivative 2.

Flashcard 44: What is the rate of change of $f(x) = 7$ ?

Answer: Rate of change is 0. Constant functions have zero rate of change.

Flashcard 45: What is the rate of change for $f(x) = x^3$ at $x = 1$ ?

Answer: Rate of change is 3. Use $f'(x) = 3x^2$ , evaluate at $x = 1$ .

Flashcard 46: Identify the derivative of $f(x) = \text{arctan}(x)$ .

Answer: $f'(x) = \frac{1}{1+x^2}$ . Standard inverse trig derivative for arctangent.

Flashcard 47: Find the rate of change of $f(x) = 4x - 7$ at $x = 2$ .

Answer: Rate of change is 4. Linear function has constant slope 4.

Flashcard 48: Find the derivative of $f(x) = \text{sec}(x)$ .

Answer: $f'(x) = \text{sec}(x)\text{tan}(x)$ . Product rule applied to $\sec(x) = \frac{1}{\cos(x)}$ .

Flashcard 49: Determine $f'(x)$ for $f(x) = \text{ln}(x)$ .

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

Flashcard 50: Determine the derivative of $f(x) = \text{arcsec}(x)$ .

Answer: $f'(x) = \frac{1}{|x|\text{sqrt}(x^2-1)}$ . Involves absolute value for proper domain.

Flashcard 51: Calculate $\frac{d}{dx}[x^5 - 2x^3]$ .

Answer: $5x^4 - 6x^2$ . Apply power rule to each term separately.

Flashcard 52: Identify the derivative of $f(x) = \text{tan}^2(x)$ .

Answer: $f'(x) = 2\text{tan}(x)\text{sec}^2(x)$ . Chain rule applied to $\tan^2(x)$ .

Flashcard 53: Determine the derivative of $f(x) = x^{\frac{1}{2}}$ .

Answer: $f'(x) = \frac{1}{2}x^{-\frac{1}{2}}$ . Power rule with fractional exponent $\frac{1}{2}$ .

Flashcard 54: Define instantaneous rate of change.

Answer: The derivative at a specific point. The slope of tangent line at one point.

Flashcard 55: Determine the derivative of $f(x) = x^{\frac{1}{2}}$ .

Answer: $f'(x) = \frac{1}{2}x^{-\frac{1}{2}}$ . Power rule with fractional exponent $\frac{1}{2}$ .

Flashcard 56: Identify the derivative of $f(x) = \tan(x)$ .

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

Flashcard 57: Determine the derivative of $f(x) = \text{arcsec}(x)$ .

Answer: $f'(x) = \frac{1}{|x|\text{sqrt}(x^2-1)}$ . Involves absolute value for proper domain.

Flashcard 58: What is the derivative of $f(x) = \text{arccsc}(x)$ ?

Answer: $f'(x) = -\frac{1}{|x|\text{sqrt}(x^2-1)}$ . Negative of arcsecant derivative.

Flashcard 59: Calculate the derivative of $f(x) = \text{e}^{2x}$ .

Answer: $f'(x) = 2\text{e}^{2x}$ . Chain rule: multiply by inner derivative 2.

Flashcard 60: Calculate the derivative of $f(x) = \text{arccot}(x)$ .

Answer: $f'(x) = -\frac{1}{1+x^2}$ . Negative of arctangent derivative.

Flashcard 61: Calculate the derivative of $f(x) = \text{arccot}(x)$ .

Answer: $f'(x) = -\frac{1}{1+x^2}$ . Negative of arctangent derivative.

Flashcard 62: Identify the derivative of $f(x) = \text{arctan}(x)$ .

Answer: $f'(x) = \frac{1}{1+x^2}$ . Standard inverse trig derivative for arctangent.

Flashcard 63: What is the rate of change for $f(x) = x^3$ at $x = 1$ ?

Answer: Rate of change is 3. Use $f'(x) = 3x^2$ , evaluate at $x = 1$ .

Flashcard 64: What is the derivative of $f(x) = \text{cos}^2(x)$ ?

Answer: $f'(x) = -2\text{sin}(x)\text{cos}(x)$ . Chain rule gives negative of $\sin^2(x)$ derivative.

Flashcard 65: Identify the derivative of $f(x) = e^x$ .

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

Flashcard 66: Determine the derivative of $f(x) = \text{cot}(x)$ .

Answer: $f'(x) = -\text{csc}^2(x)$ . Cotangent is $\frac{\cos(x)}{\sin(x)}$ , use quotient rule.

Flashcard 67: What is the derivative of $f(x) = x^2$ ?

Answer: $f'(x) = 2x$ . Apply power rule: bring down exponent 2, reduce power by 1.

Flashcard 68: What is the derivative of $f(x) = \frac{1}{x}$ ?

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

Flashcard 69: What is the derivative of $f(x) = \text{csc}(x)$ ?

Answer: $f'(x) = -\text{csc}(x)\text{cot}(x)$ . Quotient rule applied to $\csc(x) = \frac{1}{\sin(x)}$ .

AP Precalculus Flashcards: Rates Of Change

Rates Of Change

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: If y=3x3−5xy = 3x^3 - 5xy=3x3−5x, what is dydx\frac{dy}{dx}dxdy​?

Flashcard 2: What is the derivative of f(x)=log⁡a(x)f(x) = \log_a(x)f(x)=loga​(x)?

Flashcard 3: Find the derivative of f(x)=sec(x)f(x) = \text{sec}(x)f(x)=sec(x).

Flashcard 4: What is the derivative of f(x)=cos2(x)f(x) = \text{cos}^2(x)f(x)=cos2(x)?

Flashcard 5: State the derivative of f(x)=arcsin(x)f(x) = \text{arcsin}(x)f(x)=arcsin(x).

Flashcard 6: Identify the derivative of f(x)=tan2(x)f(x) = \text{tan}^2(x)f(x)=tan2(x).

Flashcard 7: Identify the derivative of f(x)=exf(x) = e^xf(x)=ex.

Flashcard 8: What is the derivative of f(x)=arccsc(x)f(x) = \text{arccsc}(x)f(x)=arccsc(x)?

Flashcard 9: State the derivative of f(x)=sin2(x)f(x) = \text{sin}^2(x)f(x)=sin2(x).

Flashcard 10: What is the rate of change of f(x)=7f(x) = 7f(x)=7?

Flashcard 11: What is the rate of change of f(x)=x4f(x) = x^4f(x)=x4 at x=2x = 2x=2?

Flashcard 12: What is the rate of change of f(x)=x2+3xf(x) = x^2 + 3xf(x)=x2+3x at x=1x=1x=1?

Flashcard 13: What is the derivative of f(x)=1x3f(x) = \frac{1}{x^3}f(x)=x31​?

Flashcard 14: What is the derivative of f(x)=ln(x2)f(x) = \text{ln}(x^2)f(x)=ln(x2)?

Flashcard 15: Identify the derivative of f(x)=tan(x)f(x) = \text{tan}(x)f(x)=tan(x).

Flashcard 16: Calculate ddx[5x2]\frac{d}{dx}[5x^2]dxd​[5x2].

Flashcard 17: State the constant rule for derivatives.

Flashcard 18: What is the derivative of f(x)=loga(x)f(x) = \text{log}_a(x)f(x)=loga​(x)?

Flashcard 19: Determine the derivative of f(x)=5xf(x) = 5^xf(x)=5x.

Flashcard 20: Find f′(x)f'(x)f′(x) for f(x)=sin(x)f(x) = \text{sin}(x)f(x)=sin(x).

Flashcard 21: What is the derivative of f(x)=arccos(x)f(x) = \text{arccos}(x)f(x)=arccos(x)?

Flashcard 22: What is the derivative of f(x)=xxf(x) = x^xf(x)=xx?

Flashcard 23: State the constant rule for derivatives.

Flashcard 24: What is the rate of change of f(x)=x2+3xf(x) = x^2 + 3xf(x)=x2+3x at x=1x=1x=1?

Flashcard 25: Determine the derivative of f(x)=5xf(x) = 5^xf(x)=5x.

Flashcard 26: What is the derivative of f(x)=cos(x)f(x) = \text{cos}(x)f(x)=cos(x)?

Flashcard 27: What is the derivative of f(x)=arccos(x)f(x) = \text{arccos}(x)f(x)=arccos(x)?

Flashcard 28: Calculate ddx[x5−2x3]\frac{d}{dx}[x^5 - 2x^3]dxd​[x5−2x3].

Flashcard 29: If y=3x3−5xy = 3x^3 - 5xy=3x3−5x, what is dydx\frac{dy}{dx}dxdy​?

Flashcard 30: Calculate ddx[5x2]\frac{d}{dx}[5x^2]dxd​[5x2].

Flashcard 31: Define average rate of change of a function f(x)f(x)f(x) over [a,b][a, b][a,b].

Flashcard 32: What is the derivative of f(x)=csc(x)f(x) = \text{csc}(x)f(x)=csc(x)?

Flashcard 33: What is the rate of change of f(x)=x4f(x) = x^4f(x)=x4 at x=2x = 2x=2?

Flashcard 34: What is the instantaneous rate of change at x=ax = ax=a?

Flashcard 35: State the derivative of f(x)=sin2(x)f(x) = \text{sin}^2(x)f(x)=sin2(x).

Flashcard 36: What is the derivative of f(x)=xxf(x) = x^xf(x)=xx?

Flashcard 37: Determine the derivative of f(x)=cot(x)f(x) = \text{cot}(x)f(x)=cot(x).

Flashcard 38: What is the derivative of f(x)=x2f(x) = x^2f(x)=x2?

Flashcard 39: What is the derivative of f(x)=ln(x2)f(x) = \text{ln}(x^2)f(x)=ln(x2)?

Flashcard 40: What is the derivative of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​?

Flashcard 41: State the derivative of f(x)=arcsin(x)f(x) = \text{arcsin}(x)f(x)=arcsin(x).