AP Calculus BC Flashcards: Introducing Calculus

What this deck covers

This deck focuses on Introducing Calculus, 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 derivative of $\text{ln}(f(x))$?

Answer: $\frac{f'(x)}{f(x)}$. Use the Chain Rule with the natural logarithm function.

Flashcard 2: What is the derivative of $\text{csc}(x)$?

Answer: $-\text{csc}(x)\text{cot}(x)$. This is the standard derivative formula for cosecant.

Flashcard 3: State the Power Rule for derivatives.

Answer: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Multiply by the exponent, then reduce the exponent by 1.

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

Answer: $f'(x) \text{e}^{f(x)}$. Use the Chain Rule with the exponential function.

Flashcard 5: Find the derivative of $f(x) = \text{sin}(2x)$.

Answer: $f'(x) = 2 \text{cos}(2x)$. Apply Chain Rule: derivative of $\sin(u)$ is $\cos(u) \cdot u'$ where $u = 2x$.

Flashcard 6: Which function's derivative is $e^x$?

Answer: $e^x$. The exponential function $e^x$ is its own derivative.

Flashcard 7: Find the derivative of $f(x) = \text{tan}(x^2)$.

Answer: $f'(x) = 2x \text{sec}^2(x^2)$. Apply Chain Rule: derivative of $\tan(u)$ is $\sec^2(u) \cdot u'$ where $u = x^2$.

Flashcard 8: State the Chain Rule for derivatives.

Answer: If $f(g(x))$, then $(f \circ g)'(x) = f'(g(x))g'(x)$. Derivative of outer function times derivative of inner function.

Flashcard 9: Identify the derivative of $f(x) = \text{cos}(3x)$.

Answer: $f'(x) = -3 \text{sin}(3x)$. Apply Chain Rule: derivative of $\cos(u)$ is $-\sin(u) \cdot u'$ where $u = 3x$.

Flashcard 10: Find the derivative of $f(x) = \frac{1}{x}$.

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

Flashcard 11: What is the derivative of $\text{arcsin}(x)$?

Answer: $\frac{1}{\sqrt{1-x^2}}$. This is the standard derivative formula for inverse sine.

Flashcard 12: Which rule is used to find the derivative of a quotient of two functions?

Answer: The Quotient Rule. Use this when differentiating quotients of two functions.

Flashcard 13: What is the derivative of $\text{sec}(x)$?

Answer: $\text{sec}(x)\text{tan}(x)$. This is the standard derivative formula for secant.

Flashcard 14: State the Product Rule for derivatives.

Answer: If $u(x)$ and $v(x)$, then $(uv)' = u'v + uv'$. First function times derivative of second plus second times derivative of first.

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

Answer: $-\text{csc}(x)\text{cot}(x)$. This is the standard derivative formula for cosecant.

Flashcard 16: Identify the derivative of $f(x) = \frac{1}{\text{sin}(x)}$.

Answer: $f'(x) = -\frac{\text{cos}(x)}{\text{sin}^2(x)}$. Rewrite as $\csc(x)$ and use the derivative formula.

Flashcard 17: Find the derivative of $f(x) = 5x^3 - 4x + 7$.

Answer: $f'(x) = 15x^2 - 4$. Apply the Power Rule to each term: $3 \cdot 5x^2 - 4 \cdot 1 + 0$.

Flashcard 18: What is the derivative of $\text{e}^{f(x)}$?

Answer: $f'(x) \text{e}^{f(x)}$. Use the Chain Rule with the exponential function.

Flashcard 19: What is the derivative of $\text{cot}(x)$?

Answer: $-\text{csc}^2(x)$. This is the standard derivative formula for cotangent.

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

Answer: $-\text{sin}(x)$. The derivative of cosine is negative sine.

Flashcard 21: Find the derivative of $f(x) = \text{ln}(x)$.

Answer: $f'(x) = \frac{1}{x}$. The natural logarithm's derivative is the reciprocal function.

Flashcard 22: Find the derivative of $f(x) = 5x^3 - 4x + 7$.

Answer: $f'(x) = 15x^2 - 4$. Apply the Power Rule to each term: $3 \cdot 5x^2 - 4 \cdot 1 + 0$.

Flashcard 23: Identify the derivative of the inverse function $f^{-1}(x)$.

Answer: $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$. The derivative of an inverse function uses this reciprocal relationship.

Flashcard 24: State the derivative of $\text{arctan}(x)$.

Answer: $\frac{1}{1+x^2}$. This is the standard derivative formula for inverse tangent.

Flashcard 25: What is the derivative of $\text{sin}(x)$?

Answer: $\text{cos}(x)$. The derivative of sine is cosine.

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

Answer: $\text{sec}(x)\text{tan}(x)$. This is the standard derivative formula for secant.

Flashcard 27: Find the derivative of $f(x) = x \text{e}^x$ using the Product Rule.

Answer: $f'(x) = \text{e}^x + x \text{e}^x$. Apply Product Rule: $1 \cdot e^x + x \cdot e^x = e^x(1 + x)$.

Flashcard 28: Identify the derivative of $f(x) = \frac{1}{\text{sin}(x)}$.

Answer: $f'(x) = -\frac{\text{cos}(x)}{\text{sin}^2(x)}$. Rewrite as $\csc(x)$ and use the derivative formula.

Flashcard 29: Find the derivative of $f(x) = 3x^2 + 2x + 1$.

Answer: $f'(x) = 6x + 2$. Apply the Power Rule to each term separately.

Flashcard 30: What is the derivative of $\text{ln}(f(x))$?

Answer: $\frac{f'(x)}{f(x)}$. Use the Chain Rule with the natural logarithm function.

Flashcard 31: Find the derivative of $f(x) = \text{ln}(x)$.

Answer: $f'(x) = \frac{1}{x}$. The natural logarithm's derivative is the reciprocal function.

Flashcard 32: Which rule is used to find the derivative of a product of two functions?

Answer: The Product Rule. Use this when differentiating products of two functions.

Flashcard 33: What is the geometric interpretation of a derivative?

Answer: The slope of the tangent line to the curve at a point. The derivative gives the instantaneous slope at any point.

Flashcard 34: What is the derivative of $x^n$ where $n$ is a constant?

Answer: $nx^{n-1}$. This is the Power Rule: bring down the exponent and subtract 1.

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

Answer: $f'(x) = 2x \text{sec}^2(x^2)$. Apply Chain Rule: derivative of $\tan(u)$ is $\sec^2(u) \cdot u'$ where $u = x^2$.

Flashcard 36: State the Quotient Rule for derivatives.

Answer: If $u(x)$ and $v(x)$, then $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$. Low times derivative of high minus high times derivative of low, over low squared.

Flashcard 37: Find the derivative of $f(x) = x^4 - 2x^2 + x$.

Answer: $f'(x) = 4x^3 - 4x + 1$. Apply Power Rule to each term: $4x^3 - 4x + 1$.

Flashcard 38: Which rule is used to find the derivative of a product of two functions?

Answer: The Product Rule. Use this when differentiating products of two functions.

Flashcard 39: What is the derivative of $\sin(x)$?

Answer: $\cos(x)$. The derivative of sine is cosine.

Flashcard 40: Which rule is used to find the derivative of a quotient of two functions?

Answer: The Quotient Rule. Use this when differentiating quotients of two functions.

Flashcard 41: Find the derivative of $f(x) = \text{ln}(x^2 + 1)$.

Answer: $f'(x) = \frac{2x}{x^2 + 1}$. Apply Chain Rule: $\frac{1}{x^2 + 1} \cdot 2x$.

Flashcard 42: Identify the derivative of the inverse function $f^{-1}(x)$.

Answer: $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$. The derivative of an inverse function uses this reciprocal relationship.

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

Answer: $f'(x) = 2\text{sin}(x)\text{cos}(x)$. Apply Chain Rule: $2\sin(x) \cdot \cos(x)$ or equivalently $\sin(2x)$.

Flashcard 44: What is the derivative of $\text{arcsin}(x)$?

Answer: $\frac{1}{\sqrt{1-x^2}}$. This is the standard derivative formula for inverse sine.

Flashcard 45: What is the derivative of $\cos(x)$?

Answer: -$\sin(x)$. The derivative of cosine is negative sine.

Flashcard 46: Find the derivative of $f(x) = \text{e}^{x^2}$.

Answer: $f'(x) = 2x \text{e}^{x^2}$. Apply Chain Rule: derivative of $e^u$ is $e^u \cdot u'$ where $u = x^2$.

Flashcard 47: What is the derivative of $\text{cot}(x)$?

Answer: $-\text{csc}^2(x)$. This is the standard derivative formula for cotangent.

Flashcard 48: State the Product Rule for derivatives.

Answer: If $u(x)$ and $v(x)$, then $(uv)' = u'v + uv'$. First function times derivative of second plus second times derivative of first.

Flashcard 49: Which rule is used for differentiating compositions of functions?

Answer: The Chain Rule. Use this rule for composite functions like $f(g(x))$.

Flashcard 50: State the derivative of $\text{arctan}(x)$.

Answer: $\frac{1}{1+x^2}$. This is the standard derivative formula for inverse tangent.

Flashcard 51: Find the derivative of $f(x) = \text{ln}(x^2 + 1)$.

Answer: $f'(x) = \frac{2x}{x^2 + 1}$. Apply Chain Rule: $\frac{1}{x^2 + 1} \cdot 2x$.

Flashcard 52: What is the derivative of $\text{arccos}(x)$?

Answer: $-\frac{1}{\sqrt{1-x^2}}$. This is the standard derivative formula for inverse cosine.

Flashcard 53: Find the derivative of $f(x) = \text{e}^{x^2}$.

Answer: $f'(x) = 2x \text{e}^{x^2}$. Apply Chain Rule: derivative of $e^u$ is $e^u \cdot u'$ where $u = x^2$.

Flashcard 54: State the Quotient Rule for derivatives.

Answer: If $u(x)$ and $v(x)$, then $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$. Low times derivative of high minus high times derivative of low, over low squared.

Flashcard 55: Identify the derivative of $f(x) = \text{sin}^2(x)$.

Answer: $f'(x) = 2\text{sin}(x)\text{cos}(x)$. Apply Chain Rule: $2\sin(x) \cdot \cos(x)$ or equivalently $\sin(2x)$.

Flashcard 56: Find the derivative of $f(x) = \frac{1}{x}$.

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

Flashcard 57: State the formula for the derivative of $f(x)$ using limits.

Answer: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. This is the formal limit definition of the derivative.

Flashcard 58: Find the derivative of $f(x) = x^4 - 2x^2 + x$.

Answer: $f'(x) = 4x^3 - 4x + 1$. Apply Power Rule to each term: $4x^3 - 4x + 1$.

Flashcard 59: Identify the derivative of a constant function.

Answer: Zero. Constants have no rate of change, so their derivative is zero.

Flashcard 60: Find the derivative of $f(x) = \text{sin}(2x)$.

Answer: $f'(x) = 2 \text{cos}(2x)$. Apply Chain Rule: derivative of $\sin(u)$ is $\cos(u) \cdot u'$ where $u = 2x$.

Flashcard 61: Find the derivative of $f(x) = 3x^2 + 2x + 1$.

Answer: $f'(x) = 6x + 2$. Apply the Power Rule to each term separately.

Flashcard 62: Find the derivative of $f(x) = x e^x$ using the Product Rule.

Answer: $f'(x) = e^x + x e^x$. Apply Product Rule: $1 \cdot e^x + x \cdot e^x = e^x(1 + x)$.

Flashcard 63: What is the geometric interpretation of a derivative?

Answer: The slope of the tangent line to the curve at a point. The derivative gives the instantaneous slope at any point.

Flashcard 64: What is the derivative of $x^n$ where $n$ is a constant?

Answer: $nx^{n-1}$. This is the Power Rule: bring down the exponent and subtract 1.

Flashcard 65: What is the derivative of $\text{tan}(x)$?

Answer: $\text{sec}^2(x)$. The derivative of tangent is secant squared.

Flashcard 66: Identify the derivative of $f(x) = \text{cos}(3x)$.

Answer: $f'(x) = -3 \text{sin}(3x)$. Apply Chain Rule: derivative of $\cos(u)$ is $-\sin(u) \cdot u'$ where $u = 3x$.

Flashcard 67: Which function's derivative is $e^x$?

Answer: $e^x$. The exponential function $e^x$ is its own derivative.

Flashcard 68: What is the derivative of $\text{arccos}(x)$?

Answer: $-\frac{1}{\sqrt{1-x^2}}$. This is the standard derivative formula for inverse cosine.

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

Answer: $\text{sec}^2(x)$. The derivative of tangent is secant squared.

Flashcard 70: State the Power Rule for derivatives.

Answer: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Multiply by the exponent, then reduce the exponent by 1.

Flashcard 71: What is the definition of a derivative?

Answer: The limit of the average rate of change as the interval approaches zero. This captures the instantaneous rate of change concept.

Flashcard 72: What is the definition of a derivative?

Answer: The limit of the average rate of change as the interval approaches zero. This captures the instantaneous rate of change concept.