AP Calculus BC Flashcards: Connecting A Function And Its Derivatives

What this deck covers

This deck focuses on Connecting A Function And Its Derivatives, 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: What is the second derivative of $f(x) = x^3$?

Answer: $f''(x) = 6x$. Apply power rule twice: $f'(x) = 3x^2$, then $f''(x) = 6x$.

Flashcard 2: Find $f''(x)$ for $f(x) = x^4$.

Answer: $f''(x) = 12x^2$. Apply power rule: $f'(x) = 4x^3$, then $f''(x) = 12x^2$.

Flashcard 3: Find $f''(x)$ for $f(x) = x^4$.

Answer: $f''(x) = 12x^2$. Apply power rule: $f'(x) = 4x^3$, then $f''(x) = 12x^2$.

Flashcard 4: Determine $f'(x)$ for $f(x) = e^x$.

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

Flashcard 5: What is the derivative of $f(x) = x \text{e}^x$?

Answer: $f'(x) = \text{e}^x + x \text{e}^x$. Apply product rule: $(uv)' = u'v + uv'$.

Flashcard 6: What can be concluded if $f''(x) < 0$ at a point?

Answer: $f(x)$ is concave down. Negative second derivative means curve bends downward.

Flashcard 7: Find $f'(x)$ for $f(x) = x^2 + 2x + 1$.

Answer: $f'(x) = 2x + 2$. Apply power rule to polynomial.

Flashcard 8: What is the derivative of $f(x) = \text{sin}(2x)$?

Answer: $f'(x) = 2\text{cos}(2x)$. Apply chain rule to $\sin(2x)$.

Flashcard 9: What does the concavity test involve?

Answer: Analyzing $f''(x)$ to determine concavity. Examining where $f''(x)$ changes sign.

Flashcard 10: Determine the inflection point of $f(x) = x^3 - 3x$.

Answer: At $x = 0$. Set $f''(x) = 6x = 0$ to find inflection points.

Flashcard 11: If $f''(x) > 0$, what can be said about $f(x)$?

Answer: $f(x)$ is concave up. Positive second derivative means curve bends upward.

Flashcard 12: Find the critical points of $f(x) = x^3 - 3x^2$.

Answer: At $x = 0$ and $x = 2$. Set $f'(x) = 3x^2 - 6x = 3x(x-2) = 0$.

Flashcard 13: Calculate $f'(x)$ for $f(x) = \tan(x)$.

Answer: $f'(x) = \sec^2(x)$. Standard derivative of tangent function.

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

Answer: $f''(x) = -\frac{2}{x^2}$. Use chain rule: $f'(x) = \frac{2}{x}$, then apply quotient rule.

Flashcard 15: Find the derivative of $f(x) = \text{e}^{-x}$.

Answer: $f'(x) = -\text{e}^{-x}$. Apply chain rule with $u = -x$.

Flashcard 16: If $f''(x) > 0$, what can be said about $f(x)$?

Answer: $f(x)$ is concave up. Positive second derivative means curve bends upward.

Flashcard 17: What is the first derivative of $f(x) = \text{ln}(x)$?

Answer: $f'(x) = \frac{1}{x}$. Standard derivative of natural logarithm function.

Flashcard 18: Determine $f''(x)$ for $f(x) = \text{cos}(2x)$.

Answer: $f''(x) = -4\text{cos}(2x)$. Apply chain rule: derivative of $\cos(2x)$ is $-2\sin(2x)$, then again.

Flashcard 19: Identify 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 20: What is the derivative of $f(x) = \text{cos}(x)$?

Answer: $f'(x) = -\text{sin}(x)$. Standard derivative of cosine function.

Flashcard 21: What does the second derivative of a function represent?

Answer: The concavity of the function. Second derivative determines curve shape and bending.

Flashcard 22: What does the concavity test involve?

Answer: Analyzing $f''(x)$ to determine concavity. Examining where $f''(x)$ changes sign.

Flashcard 23: Identify the critical points of $f(x) = x^2 - 4x + 4$.

Answer: At $x = 2$. Set $f'(x) = 2x - 4 = 0$ to find critical points.

Flashcard 24: What can be concluded if $f''(x) < 0$ at a point?

Answer: $f(x)$ is concave down. Negative second derivative means curve bends downward.

Flashcard 25: Determine $f'(x)$ for $f(x) = \text{ln}(2x)$.

Answer: $f'(x) = \frac{1}{x}$. Chain rule: derivative of $\ln(2x)$ is $\frac{2}{2x} = \frac{1}{x}$.

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

Answer: $f''(x) = -\text{cos}(x)$. Derivative of cosine is $-\sin(x)$; derivative again gives $-\cos(x)$.

Flashcard 27: What is the second derivative of $f(x) = \text{ln}(x^2)$?

Answer: $f''(x) = -\frac{2}{x^2}$. Use chain rule: $f'(x) = \frac{2}{x}$, then apply quotient rule.

Flashcard 28: Determine the second derivative of $f(x) = \text{tan}(x)$.

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

Flashcard 29: Find $f'(x)$ for $f(x) = \frac{1}{x^2}$.

Answer: $f'(x) = -\frac{2}{x^3}$. Rewrite as $x^{-2}$ and apply power rule.

Flashcard 30: What is the second derivative of $f(x) = x^5$?

Answer: $f''(x) = 20x^3$. Apply power rule: $f'(x) = 5x^4$, then $f''(x) = 20x^3$.

Flashcard 31: What does the first derivative of a function represent?

Answer: The slope of the tangent line to the curve at a point. Derivative measures instantaneous rate of change.

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

Answer: $f''(x) = \text{e}^x$. Exponential function equals its own second derivative.

Flashcard 33: What is the first derivative test used for?

Answer: To find intervals of increase or decrease. Uses sign of $f'(x)$ to determine monotonicity.

Flashcard 34: What is the first derivative of $f(x) = \text{sin}^2(x)$?

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

Flashcard 35: Find the critical points of $f(x) = x^3 - 3x^2$.

Answer: At $x = 0$ and $x = 2$. Set $f'(x) = 3x^2 - 6x = 3x(x-2) = 0$.

Flashcard 36: What is the second derivative test used for?

Answer: To determine if a critical point is a local max or min. Uses sign of $f''(c)$ to classify critical points.

Flashcard 37: Find $f''(x)$ for $f(x) = \frac{1}{x}$.

Answer: $f''(x) = \frac{2}{x^3}$. Rewrite as $x^{-1}$, apply power rule twice.

Flashcard 38: What does $f'(x) = 0$ indicate about $f(x)$ at $x$?

Answer: Possible local maxima, minima, or inflection point. Zero derivative indicates horizontal tangent line.

Flashcard 39: Determine $f'(x)$ for $f(x) = \text{ln}(2x)$.

Answer: $f'(x) = \frac{1}{x}$. Chain rule: derivative of $\ln(2x)$ is $\frac{2}{2x} = \frac{1}{x}$.

Flashcard 40: Determine the first derivative of $f(x) = \text{e}^{2x}$.

Answer: $f'(x) = 2\text{e}^{2x}$. Apply chain rule: $\frac{d}{dx}[e^{u}] = e^{u} \cdot u'$.

Flashcard 41: Determine $f'(x)$ for $f(x) = e^x$.

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

Flashcard 42: Find the derivative of $f(x) = \text{e}^{-x}$.

Answer: $f'(x) = -\text{e}^{-x}$. Apply chain rule with $u = -x$.

Flashcard 43: Identify 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 44: What is the second derivative of $f(x) = \text{e}^x$?

Answer: $f''(x) = \text{e}^x$. Exponential function equals its own second derivative.

Flashcard 45: Find $f'(x)$ for $f(x) = \frac{1}{x^2}$.

Answer: $f'(x) = -\frac{2}{x^3}$. Rewrite as $x^{-2}$ and apply power rule.

Flashcard 46: What does the first derivative of a function represent?

Answer: The slope of the tangent line to the curve at a point. Derivative measures instantaneous rate of change.

Flashcard 47: Determine the inflection point of $f(x) = x^3 - 3x$.

Answer: At $x = 0$. Set $f''(x) = 6x = 0$ to find inflection points.

Flashcard 48: Find $f''(x)$ for $f(x) = \frac{1}{x}$.

Answer: $f''(x) = \frac{2}{x^3}$. Rewrite as $x^{-1}$, apply power rule twice.

Flashcard 49: What is the derivative of $f(x) = \text{cos}(x)$?

Answer: $f'(x) = -\text{sin}(x)$. Standard derivative of cosine function.

Flashcard 50: What is the first derivative of $f(x) = \text{sin}^2(x)$?

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

Flashcard 51: What is the second derivative of $f(x) = \ln(x)$?

Answer: $f''(x) = -\frac{1}{x^2}$. Derivative of $\frac{1}{x}$ applied to $\ln(x)$.

Flashcard 52: What is the second derivative test used for?

Answer: To determine if a critical point is a local max or min. Uses sign of $f''(c)$ to classify critical points.

Flashcard 53: State the power rule for differentiation.

Answer: $\frac{d}{dx}[x^n] = nx^{n-1}$. Fundamental rule for differentiating polynomial terms.

Flashcard 54: What is the first derivative of $f(x) = \ln(x)$?

Answer: $f'(x) = \frac{1}{x}$. Standard derivative of natural logarithm function.

Flashcard 55: What does $f'(x) = 0$ indicate about $f(x)$ at $x$?

Answer: Possible local maxima, minima, or inflection point. Zero derivative indicates horizontal tangent line.

Flashcard 56: What is the derivative of $f(x) = x \text{e}^x$?

Answer: $f'(x) = \text{e}^x + x \text{e}^x$. Apply product rule: $(uv)' = u'v + uv'$.

Flashcard 57: Identify the derivative of $f(x) = \frac{1}{x}$.

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

Flashcard 58: Find $f'(x)$ for $f(x) = x^2 + 2x + 1$.

Answer: $f'(x) = 2x + 2$. Apply power rule to polynomial.

Flashcard 59: Determine $f''(x)$ for $f(x) = \text{cos}(2x)$.

Answer: $f''(x) = -4\text{cos}(2x)$. Apply chain rule: derivative of $\cos(2x)$ is $-2\sin(2x)$, then again.

Flashcard 60: Identify the derivative of $f(x) = \frac{1}{x}$.

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

Flashcard 61: What is the second derivative of $f(x) = \text{ln}(x)$?

Answer: $f''(x) = -\frac{1}{x^2}$. Derivative of $\frac{1}{x}$ applied to $\ln(x)$.

Flashcard 62: What is the second derivative of $f(x) = \text{cos}(x)$?

Answer: $f''(x) = -\text{cos}(x)$. Derivative of cosine is $-\sin(x)$; derivative again gives $-\cos(x)$.

Flashcard 63: What does the second derivative of a function represent?

Answer: The concavity of the function. Second derivative determines curve shape and bending.

Flashcard 64: What is the second derivative of $f(x) = x^5$?

Answer: $f''(x) = 20x^3$. Apply power rule: $f'(x) = 5x^4$, then $f''(x) = 20x^3$.

Flashcard 65: What is the derivative of $f(x) = x^2$?

Answer: $f'(x) = 2x$. Apply power rule: $\frac{d}{dx}[x^n] = nx^{n-1}$.

Flashcard 66: Determine the first derivative of $f(x) = \text{e}^{2x}$.

Answer: $f'(x) = 2\text{e}^{2x}$. Apply chain rule: $\frac{d}{dx}[e^{u}] = e^{u} \cdot u'$.

Flashcard 67: Identify the critical points of $f(x) = x^2 - 4x + 4$.

Answer: At $x = 2$. Set $f'(x) = 2x - 4 = 0$ to find critical points.

Flashcard 68: Calculate $f'(x)$ for $f(x) = \text{tan}(x)$.

Answer: $f'(x) = \text{sec}^2(x)$. Standard derivative of tangent function.

Flashcard 69: State the power rule for differentiation.

Answer: $\frac{d}{dx}[x^n] = nx^{n-1}$. Fundamental rule for differentiating polynomial terms.

Flashcard 70: Calculate $f''(x)$ for $f(x) = \text{sin}(x)$.

Answer: $f''(x) = -\text{sin}(x)$. Derivative of sine is cosine; derivative of cosine is $-\sin(x)$.

Flashcard 71: What is the derivative of $f(x) = x^2$?

Answer: $f'(x) = 2x$. Apply power rule: $\frac{d}{dx}[x^n] = nx^{n-1}$.

Flashcard 72: What is the first derivative test used for?

Answer: To find intervals of increase or decrease. Uses sign of $f'(x)$ to determine monotonicity.

Flashcard 73: Determine the second derivative of $f(x) = \text{tan}(x)$.

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

Flashcard 74: What is the second derivative of $f(x) = x^3$?

Answer: $f''(x) = 6x$. Apply power rule twice: $f'(x) = 3x^2$, then $f''(x) = 6x$.

Flashcard 75: Calculate $f''(x)$ for $f(x) = \text{sin}(x)$.

Answer: $f''(x) = -\text{sin}(x)$. Derivative of sine is cosine; derivative of cosine is $-\sin(x)$.

Flashcard 76: What is the derivative of $f(x) = \text{sin}(2x)$?

Answer: $f'(x) = 2\text{cos}(2x)$. Apply chain rule to $\sin(2x)$.