AP Precalculus Flashcards: Rates Of Change

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.

Flashcard 1: 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 2: Identify the derivative of $f(x) = e^x$.

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

Flashcard 3: 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 4: What is the rate of change of $f(x) = 7$?

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

Flashcard 5: 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 6: 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 7: 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 8: Identify the derivative of $f(x) = \text{tan}(x)$.

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

Flashcard 9: 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 10: Find $f'(x)$ for $f(x) = \text{sin}(x)$.

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

Flashcard 11: 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 12: 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 13: State the constant rule for derivatives.

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

Flashcard 14: 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 15: What is the derivative of $f(x) = \text{cos}(x)$?

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

Flashcard 16: Calculate $\frac{d}{dx}[x^5 - 2x^3]$.

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

Flashcard 17: 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 18: Calculate $\frac{d}{dx}[5x^2]$.

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

Flashcard 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: 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 26: State the power rule for derivatives.

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

Flashcard 27: Calculate the derivative of $f(x) = \text{e}^{2x}$.

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

Flashcard 28: Identify the derivative of $f(x) = \text{arctan}(x)$.

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

Flashcard 29: 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 30: 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 31: Determine $f'(x)$ for $f(x) = \text{ln}(x)$.

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

Flashcard 32: 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 33: 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 34: 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 35: Define instantaneous rate of change.

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

Flashcard 36: Identify the derivative of $f(x) = \tan(x)$.

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

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

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

Flashcard 38: 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 39: 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 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: What is the derivative of $f(x) = \text{csc}(x)$?

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