AP Calculus AB Flashcards: Introduction To Related Rates

What this deck covers

This deck focuses on Introduction To Related Rates, 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: Determine $\frac{dA}{dt}$ for $A = s^2$ if $s = 10$ and $\frac{ds}{dt} = 2$.

Answer: $\frac{dA}{dt} = 40$. Square area formula: $\frac{dA}{dt} = 2s \cdot 2 = 40$.

Flashcard 2: Determine $\frac{dC}{dt}$ if $C = 2\text{π}r$ and $\frac{dr}{dt} = 5$ at $r = 4$.

Answer: $\frac{dC}{dt} = 10\text{π}$. Circumference formula differentiated.

Flashcard 3: Identify the formula for the derivative of $\text{sin}(x)$.

Answer: $\frac{d}{dx}(\text{sin}(x)) = \text{cos}(x)$. Sine derivative is cosine.

Flashcard 4: What is the derivative of $\frac{d}{dx}(\text{ln}(kx))$?

Answer: $\frac{d}{dx}(\text{ln}(kx)) = \frac{1}{x}$. Chain rule cancels constant $k$.

Flashcard 5: What is the derivative of $\frac{d}{dx}(\text{ln}(x))$?

Answer: $\frac{d}{dx}(\text{ln}(x)) = \frac{1}{x}$. Natural logarithm derivative formula.

Flashcard 6: Find $\frac{dV}{dt}$ if $V = \frac{4}{3}\times \pi r^3$ and $\frac{dr}{dt} = 2$ at $r = 3$.

Answer: $\frac{dV}{dt} = 72\pi$. Volume formula for sphere, differentiated.

Flashcard 7: Determine $\frac{dC}{dt}$ for $C = 2\pi r$ if $r = 10$ and $\frac{dr}{dt} = 0.1$.

Answer: $\frac{dC}{dt} = 0.2\pi$. Circumference: $\frac{dC}{dt} = 2\pi \cdot 0.1 = 0.2\pi$.

Flashcard 8: Determine $\frac{dA}{dt}$ for $A = s^2$ if $s = 10$ and $\frac{ds}{dt} = 2$.

Answer: $\frac{dA}{dt} = 40$. Square area formula: $\frac{dA}{dt} = 2s \cdot 2 = 40$.

Flashcard 9: What is the relationship between rates of change in related rates problems?

Answer: They are connected by the chain rule. Chain rule connects changing variables.

Flashcard 10: Identify the formula for the derivative of $\text{sin}(x)$.

Answer: $\frac{d}{dx}(\text{sin}(x)) = \text{cos}(x)$. Sine derivative is cosine.

Flashcard 11: What is the derivative of $\frac{d}{dx}(\text{arccos}(x))$?

Answer: $\frac{d}{dx}(\text{arccos}(x)) = -\frac{1}{\text{sqrt}(1-x^2)}$. Inverse cosine derivative is negative.

Flashcard 12: What is the basic strategy for solving related rates problems?

Answer: Identify variables, relate them, differentiate, solve. Standard approach for related rates.

Flashcard 13: What is the relationship between rates of change in related rates problems?

Answer: They are connected by the chain rule. Chain rule connects changing variables.

Flashcard 14: State the formula for the derivative of $\frac{d}{dx}(e^x)$.

Answer: $\frac{d}{dx}(e^x) = e^x$. Exponential function derivative equals itself.

Flashcard 15: What is the derivative of $\frac{d}{dx}(\text{csc}(x))$?

Answer: $\frac{d}{dx}(\text{csc}(x)) = -\text{csc}(x)\text{cot}(x)$. Cosecant derivative is negative.

Flashcard 16: Which rule is primarily used in related rates problems?

Answer: Chain rule. Links rates through composite functions.

Flashcard 17: What is the derivative of $\frac{d}{dx}(\text{e}^{kx})$?

Answer: $\frac{d}{dx}(\text{e}^{kx}) = k\text{e}^{kx}$. Chain rule with exponential function.

Flashcard 18: Calculate $\frac{dh}{dt}$ for $h = \text{tan}(t)$ when $t = \frac{\text{π}}{4}$.

Answer: $\frac{dh}{dt} = 2$. At $t = \frac{\pi}{4}$, $\sec^2(\frac{\pi}{4}) = 2$.

Flashcard 19: What is the derivative of $\frac{d}{dx}(\text{arctan}(x))$?

Answer: $\frac{d}{dx}(\text{arctan}(x)) = \frac{1}{1+x^2}$. Inverse tangent derivative formula.

Flashcard 20: Calculate $\frac{dA}{dt}$ for $A = \pi r^2$ if $\frac{dr}{dt} = 1$ and $r = 5$.

Answer: $\frac{dA}{dt} = 10\pi$. Circle area formula differentiated.

Flashcard 21: Determine $\frac{dy}{dt}$ for $y = x^2$ if $\frac{dx}{dt} = 4$ and $x = 3$.

Answer: $\frac{dy}{dt} = 24$. Apply chain rule: $\frac{dy}{dt} = 2x \cdot 4 = 24$.

Flashcard 22: What is the derivative of $\frac{d}{dx}(\text{ln}(kx))$?

Answer: $\frac{d}{dx}(\text{ln}(kx)) = \frac{1}{x}$. Chain rule cancels constant $k$.

Flashcard 23: What is the derivative of $\frac{d}{dx}(\text{arccos}(x))$?

Answer: $\frac{d}{dx}(\text{arccos}(x)) = -\frac{1}{\text{sqrt}(1-x^2)}$. Inverse cosine derivative is negative.

Flashcard 24: Calculate $\frac{dh}{dt}$ for $h = \text{tan}(t)$ when $t = \frac{\text{π}}{4}$.

Answer: $\frac{dh}{dt} = 2$. At $t = \frac{\pi}{4}$, $\sec^2(\frac{\pi}{4}) = 2$.

Flashcard 25: What is the derivative of $\frac{d}{dx}(\text{log}_a{x})$?

Answer: $\frac{d}{dx}(\text{log}_a{x}) = \frac{1}{x\text{ln}(a)}$. Logarithm derivative with base $a$.

Flashcard 26: What is the derivative of $\frac{d}{dx}(\text{sec}(x))$?

Answer: $\frac{d}{dx}(\text{sec}(x)) = \text{sec}(x)\text{tan}(x)$. Secant derivative involves secant and tangent.

Flashcard 27: What is the derivative of $\frac{d}{dx}(\text{ln}(x))$?

Answer: $\frac{d}{dx}(\text{ln}(x)) = \frac{1}{x}$. Natural logarithm derivative formula.

Flashcard 28: What is the derivative of $\frac{d}{dx}(\text{e}^{kx})$?

Answer: $\frac{d}{dx}(\text{e}^{kx}) = k\text{e}^{kx}$. Chain rule with exponential function.

Flashcard 29: What is the derivative of $\frac{d}{dx}(\text{log}_a{x})$?

Answer: $\frac{d}{dx}(\text{log}_a{x}) = \frac{1}{x\text{ln}(a)}$. Logarithm derivative with base $a$.

Flashcard 30: What is the formula for the derivative of $\frac{f}{g}$?

Answer: $(f/g)' = \frac{f'g - fg'}{g^2}$. Quotient rule for derivatives.

Flashcard 31: State the formula for the derivative of $\frac{d}{dx}(e^x)$.

Answer: $\frac{d}{dx}(e^x) = e^x$. Exponential function derivative equals itself.

Flashcard 32: Determine $\frac{dC}{dt}$ if $C = 2\text{π}r$ and $\frac{dr}{dt} = 5$ at $r = 4$.

Answer: $\frac{dC}{dt} = 10\text{π}$. Circumference formula differentiated.

Flashcard 33: Determine $\frac{dy}{dt}$ for $y = x^3$ if $x = 2$ and $\frac{dx}{dt} = 3$.

Answer: $\frac{dy}{dt} = 36$. Power rule: $\frac{dy}{dt} = 3x^2 \cdot 3 = 36$.

Flashcard 34: What is the formula for the derivative of $\frac{f}{g}$?

Answer: $(f/g)' = \frac{f'g - fg'}{g^2}$. Quotient rule for derivatives.

Flashcard 35: Determine $\frac{dy}{dt}$ for $y = x^2$ if $\frac{dx}{dt} = 4$ and $x = 3$.

Answer: $\frac{dy}{dt} = 24$. Apply chain rule: $\frac{dy}{dt} = 2x \cdot 4 = 24$.

Flashcard 36: Find $\frac{dV}{dt}$ if $V = \frac{4}{3}\times \pi r^3$ and $\frac{dr}{dt} = 2$ at $r = 3$.

Answer: $\frac{dV}{dt} = 72\pi$. Volume formula for sphere, differentiated.

Flashcard 37: What is the chain rule for finding derivatives?

Answer: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Composes derivatives for nested functions.

Flashcard 38: What is the formula for the derivative of $\frac{d}{dx}(\cos(x))$?

Answer: $\frac{d}{dx}(\cos(x)) = -\sin(x)$. Cosine derivative is negative sine.

Flashcard 39: What is the derivative of $\frac{d}{dx}(a^x)$?

Answer: $\frac{d}{dx}(a^x) = a^x\text{ln}(a)$. General exponential function derivative.

Flashcard 40: What is the derivative of $\frac{d}{dx}(\text{csc}(x))$?

Answer: $\frac{d}{dx}(\text{csc}(x)) = -\text{csc}(x)\text{cot}(x)$. Cosecant derivative is negative.

Flashcard 41: What is the derivative of $\frac{d}{dx}(\text{cot}(x))$?

Answer: $\frac{d}{dx}(\text{cot}(x)) = -\text{csc}^2(x)$. Cotangent derivative uses cosecant squared.

Flashcard 42: Which rule is primarily used in related rates problems?

Answer: Chain rule. Links rates through composite functions.

Flashcard 43: What is the derivative of $\frac{d}{dx}(\frac{1}{x})$?

Answer: $\frac{d}{dx}(\frac{1}{x}) = -\frac{1}{x^2}$. Power rule with $n = -1$.

Flashcard 44: State the formula for the derivative of a product of two functions.

Answer: $(f \times g)' = f' \times g + f \times g'$. Product rule for derivatives.

Flashcard 45: What is the derivative of $\frac{d}{dx}(\text{sec}(x))$?

Answer: $\frac{d}{dx}(\text{sec}(x)) = \text{sec}(x)\text{tan}(x)$. Secant derivative involves secant and tangent.

Flashcard 46: What is the derivative of $\frac{d}{dx}(\text{cot}(x))$?

Answer: $\frac{d}{dx}(\text{cot}(x)) = -\text{csc}^2(x)$. Cotangent derivative uses cosecant squared.

Flashcard 47: What is the derivative of $\frac{d}{dx}(\text{arctan}(x))$?

Answer: $\frac{d}{dx}(\text{arctan}(x)) = \frac{1}{1+x^2}$. Inverse tangent derivative formula.

Flashcard 48: What is the chain rule for finding derivatives?

Answer: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Composes derivatives for nested functions.

Flashcard 49: Find $\frac{dV}{dt}$ for $V = \frac{4}{3}\text{π}r^3$ if $r = 6$ and $\frac{dr}{dt} = 0.5$.

Answer: $\frac{dV}{dt} = 72\text{π}$. Sphere volume: $\frac{dV}{dt} = 4\pi r^2 \cdot 0.5 = 72\pi$.

Flashcard 50: What is the derivative of $\frac{d}{dx}(\text{arcsin}(x))$?

Answer: $\frac{d}{dx}(\text{arcsin}(x)) = \frac{1}{\text{sqrt}(1-x^2)}$. Inverse sine derivative formula.

Flashcard 51: Determine $\frac{dy}{dt}$ for $y = x^3$ if $x = 2$ and $\frac{dx}{dt} = 3$.

Answer: $\frac{dy}{dt} = 36$. Power rule: $\frac{dy}{dt} = 3x^2 \cdot 3 = 36$.

Flashcard 52: What is the derivative of $\frac{d}{dx}(\frac{1}{x})$?

Answer: $\frac{d}{dx}(\frac{1}{x}) = -\frac{1}{x^2}$. Power rule with $n = -1$.

Flashcard 53: Calculate $\frac{dA}{dt}$ for $A = \pi r^2$ if $r = 7$ and $\frac{dr}{dt} = 0.5$.

Answer: $\frac{dA}{dt} = 7\pi$. Circle area: $\frac{dA}{dt} = 2\pi r \cdot 0.5 = 7\pi$.

Flashcard 54: Calculate $\frac{dA}{dt}$ for $A = \pi r^2$ if $r = 7$ and $\frac{dr}{dt} = 0.5$.

Answer: $\frac{dA}{dt} = 7\pi$. Circle area: $\frac{dA}{dt} = 2\pi r \cdot 0.5 = 7\pi$.

Flashcard 55: What is the formula for the derivative of $\frac{d}{dx}(\text{cos}(x))$?

Answer: $\frac{d}{dx}(\text{cos}(x)) = -\text{sin}(x)$. Cosine derivative is negative sine.

Flashcard 56: Calculate $\frac{dA}{dt}$ for $A = \pi r^2$ if $\frac{dr}{dt} = 1$ and $r = 5$.

Answer: $\frac{dA}{dt} = 10\pi$. Circle area formula differentiated.

Flashcard 57: Find $\frac{dV}{dt}$ for $V = \frac{4}{3} \pi r^3$ if $r = 6$ and $\frac{dr}{dt} = 0.5$.

Answer: $\frac{dV}{dt} = 72\pi$. Sphere volume: $\frac{dV}{dt} = 4\pi r^2 \cdot 0.5 = 72\pi$.

Flashcard 58: What is the derivative of $\frac{d}{dx}(\text{arcsin}(x))$?

Answer: $\frac{d}{dx}(\text{arcsin}(x)) = \frac{1}{\text{sqrt}(1-x^2)}$. Inverse sine derivative formula.

Flashcard 59: Identify the formula for the derivative of $x^n$.

Answer: $\frac{d}{dx}(x^n) = nx^{n-1}$. Power rule for polynomial derivatives.

Flashcard 60: What is the basic strategy for solving related rates problems?

Answer: Identify variables, relate them, differentiate, solve. Standard approach for related rates.

Flashcard 61: State the formula for the derivative of a product of two functions.

Answer: $(f \times g)' = f' \times g + f \times g'$. Product rule for derivatives.

Flashcard 62: Determine $\frac{dC}{dt}$ for $C = 2\text{π}r$ if $r = 10$ and $\frac{dr}{dt} = 0.1$.

Answer: $\frac{dC}{dt} = 0.2\text{π}$. Circumference: $\frac{dC}{dt} = 2\pi \cdot 0.1 = 0.2\pi$.

Flashcard 63: Identify the formula for the derivative of $x^n$.

Answer: $\frac{d}{dx}(x^n) = nx^{n-1}$. Power rule for polynomial derivatives.