AP Calculus BC Flashcards: Rates Of Change In Applied Concepts

What this deck covers

This deck focuses on Rates Of Change In Applied Concepts, 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: Determine the derivative of $f(x) = \cos(x)$.

Answer: $f'(x) = -\sin(x)$. Cosine derivative is negative sine.

Flashcard 2: Determine $\frac{d}{dx}(5x - 1)$ at $x = 1$.

Answer: $5$, so the rate is $5$. Linear function has constant derivative.

Flashcard 3: Find the rate of change of the volume of a sphere with respect to its radius.

Answer: $\frac{dV}{dr} = 4\pi r^2$. Derivative of $V = \frac{4}{3}\pi r^3$ using power rule.

Flashcard 4: Find the derivative of $f(x) = \arcsin(x)$.

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

Flashcard 5: Determine $\frac{d}{dx}(x^2 + 3x + 2)$.

Answer: $2x + 3$. Sum rule and power rule applied term by term.

Flashcard 6: What is the derivative of $f(x) = \cot(x)$?

Answer: $f'(x) = -\csc^2(x)$. Cotangent derivative is negative cosecant squared.

Flashcard 7: What is the rate of change of the area of a circle with respect to its radius?

Answer: $\frac{dA}{dr} = 2\pi r$. Derivative of $A = \pi r^2$ using power rule.

Flashcard 8: Calculate the derivative of $f(x) = \sin(x)$.

Answer: $f'(x) = \cos(x)$. Sine derivative is cosine.

Flashcard 9: Evaluate $\frac{d}{dx}(x^2 + x)$ at $x = 2$.

Answer: $2x + 1$, so $f'(2) = 5$. Sum rule applied, then evaluate at $x = 2$.

Flashcard 10: What is the derivative of $f(x) = \tan(x)$?

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

Flashcard 11: State the product rule for the derivative of $u(x)v(x)$.

Answer: $(uv)' = u'v + uv'$. Product rule for multiplied functions.

Flashcard 12: Find the derivative of $f(x) = \sec(x)$.

Answer: $f'(x) = \sec(x)\tan(x)$. Secant derivative formula.

Flashcard 13: Calculate the derivative of $f(x) = \ln(ax)$ where $a$ is a constant.

Answer: $f'(x) = \frac{1}{x}$. Constant factor $a$ cancels in logarithm derivative.

Flashcard 14: What is the rate of change of $y = 2x^3$ at $x = 2$?

Answer: $f'(x) = 6x^2$, so $f'(2) = 24$. Constant multiple rule with power rule.

Flashcard 15: Find the rate of change of $f(x) = x^3$ at $x = 4$.

Answer: $f'(x) = 3x^2$, so $f'(4) = 48$. Power rule gives $3x^2$, substitute $x = 4$.

Flashcard 16: Calculate the derivative of $f(x) = 2x^3 - 3x^2 + x$.

Answer: $6x^2 - 6x + 1$. Power rule applied to polynomial terms.

Flashcard 17: Find the rate of change for $f(x) = 4x^2$ at $x = 3$.

Answer: $f'(x) = 8x$, so $f'(3) = 24$. Power rule gives $8x$, substitute $x = 3$.

Flashcard 18: Evaluate $\frac{d}{dx}(x^3 - 2x + 4)$ at $x = 0$.

Answer: $3x^2 - 2$, so $f'(0) = -2$. Differentiate then evaluate at $x = 0$.

Flashcard 19: Calculate the derivative of $f(x) = 5x^2 - 3x + 7$.

Answer: $10x - 3$. Power rule applied to polynomial.

Flashcard 20: Identify the rate of change of $y = x^3$ at $x = 2$.

Answer: $f'(x) = 3x^2$, so $f'(2) = 12$. Apply power rule then substitute $x = 2$.

Flashcard 21: Find the derivative of $f(x) = e^x$.

Answer: $f'(x) = e^x$. Exponential function derivative equals itself.

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

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

Flashcard 23: Determine the derivative of $f(x) = \arctan(x)$.

Answer: $f'(x) = \frac{1}{1+x^2}$. Arctangent derivative formula.

Flashcard 24: State the Chain Rule formula for derivatives.

Answer: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Composite function differentiation rule.

Flashcard 25: What is the formula for the rate of change of a function $f(x)$?

Answer: $f'(x)$. The derivative represents instantaneous rate of change.

Flashcard 26: Calculate the rate of change of $f(x) = x^2 + 2x$ at $x = 1$.

Answer: $f'(x) = 2x + 2$, so $f'(1) = 4$. Apply power and sum rules, then evaluate.

Flashcard 27: Determine the rate of change for $f(x) = x^2$ at $x = 5$.

Answer: $f'(x) = 2x$, so $f'(5) = 10$. Power rule gives $2x$, substitute $x = 5$.

Flashcard 28: Calculate the derivative of $f(x) = x^{-1}$.

Answer: $f'(x) = -x^{-2}$. Power rule applied to negative exponent.

Flashcard 29: Find $\frac{d}{dx} (3x^3 - 5x^2 + 4)$.

Answer: $9x^2 - 10x$. Power rule applied to each term.

Flashcard 30: Evaluate the derivative of $f(x) = \frac{1}{x}$ at $x = 1$.

Answer: $-\frac{1}{x^2}$, so $f'(1) = -1$. Reciprocal function derivative using power rule.