AP Calculus BC Flashcards: Derivatives Of Trigonometry And Logarithmic Functions

What this deck covers

This deck focuses on Derivatives Of Trigonometry And Logarithmic Functions, 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: Evaluate $f'(x)$ for $f(x) = \text{cos } x - e^x$ at $x = 0$.

Answer: $f'(0) = -1$. $f'(x) = -\text{sin } x - e^x$, so $f'(0) = 0 - 1 = -1$.

Flashcard 2: Find the second derivative of $y = \text{sin } x$.

Answer: $\frac{d^2y}{dx^2} = -\text{sin } x$. Differentiate $f'(x) = \text{cos } x$ to get $f''(x) = -\text{sin } x$.

Flashcard 3: Find the derivative: $y = \text{ln } x$.

Answer: $\frac{dy}{dx} = \frac{1}{x}$. Apply the derivative rule for $\text{ln } x$.

Flashcard 4: Find the derivative of $f(x) = e^{2x}$.

Answer: $f'(x) = 2e^{2x}$. Use chain rule with inner function $2x$.

Flashcard 5: Evaluate the derivative: $f(x) = \text{ln } x$ at $x = 2$.

Answer: $f'(2) = \frac{1}{2}$. $f'(x) = \frac{1}{x}$, so $f'(2) = \frac{1}{2}$.

Flashcard 6: Find the second derivative of $y = \text{ln } x$.

Answer: $\frac{d^2y}{dx^2} = -\frac{1}{x^2}$. Differentiate $f'(x) = \frac{1}{x}$ to get $f''(x) = -\frac{1}{x^2}$.

Flashcard 7: Evaluate the derivative: $f(x) = \text{ln } x$ at $x = 1$.

Answer: $f'(1) = 1$. $f'(x) = \frac{1}{x}$, so $f'(1) = \frac{1}{1} = 1$.

Flashcard 8: Differentiate $f(x) = e^x$ at $x = 1$.

Answer: $f'(1) = e$. $f'(x) = e^x$, so $f'(1) = e^1 = e$.

Flashcard 9: Find the derivative of $f(x) = \text{ln }(5x)$.

Answer: $f'(x) = \frac{1}{x}$. Constant multiple rule: $\frac{d}{dx}[\text{ln}(5x)] = \frac{1}{5x} \cdot 5 = \frac{1}{x}$.

Flashcard 10: Evaluate the derivative: $f(x) = \text{sin } x$ at $x = 0$.

Answer: $f'(0) = 1$. $f'(x) = \text{cos } x$, so $f'(0) = \text{cos}(0) = 1$.

Flashcard 11: Find the second derivative of $y = e^x$.

Answer: $\frac{d^2y}{dx^2} = e^x$. Differentiate $f'(x) = e^x$ to get $f''(x) = e^x$.

Flashcard 12: Determine the derivative of $f(x) = xe^x$.

Answer: $f'(x) = e^x + xe^x$. Use product rule: $(uv)' = u'v + uv'$.

Flashcard 13: Determine $f'(x)$ if $f(x) = \text{cos } x + \text{sin } x$.

Answer: $f'(x) = -\text{sin } x + \text{cos } x$. Use sum rule: derivative of each term separately.

Flashcard 14: Differentiate $f(x) = \text{sin } x$ at $x = \frac{\text{π}}{2}$.

Answer: $f'\big(\frac{\text{π}}{2}\big) = 0$. $f'(x) = \text{cos } x$, so $f'\big(\frac{\text{π}}{2}\big) = \text{cos}\big(\frac{\text{π}}{2}\big) = 0$.

Flashcard 15: Differentiate $f(x) = \text{sin } x$ at $x = \frac{\text{π}}{2}$.

Answer: $f'\big(\frac{\text{π}}{2}\big) = 0$. $f'(x) = \text{cos } x$, so $f'\big(\frac{\text{π}}{2}\big) = \text{cos}\big(\frac{\text{π}}{2}\big) = 0$.

Flashcard 16: Evaluate the derivative: $f(x) = \text{cos } x$ at $x = \frac{\text{π}}{2}$.

Answer: $f'\big(\frac{\text{π}}{2}\big) = -1$. $f'(x) = -\text{sin } x$, so $f'\big(\frac{\text{π}}{2}\big) = -\text{sin}\big(\frac{\text{π}}{2}\big) = -1$.

Flashcard 17: What is the derivative of $\text{sin } x$?

Answer: $\text{cos } x$. Basic derivative rule for sine function.

Flashcard 18: Find the derivative of $f(x) = e^{2x}$.

Answer: $f'(x) = 2e^{2x}$. Use chain rule with inner function $2x$.

Flashcard 19: Find the derivative of $f(x) = \text{cos }(3x)$.

Answer: $f'(x) = -3\text{sin }(3x)$. Use chain rule with inner function $3x$.

Flashcard 20: Find the derivative: $y = \text{cos } x$.

Answer: $\frac{dy}{dx} = -\text{sin } x$. Apply the derivative rule for $\text{cos } x$.

Flashcard 21: Find the slope of the tangent to $y = \text{ln } x$ at $x = 4$.

Answer: Slope = $\frac{1}{4}$. The slope equals the derivative at the point.

Flashcard 22: Find the derivative: $y = \text{ln } x$.

Answer: $\frac{dy}{dx} = \frac{1}{x}$. Apply the derivative rule for $\text{ln } x$.

Flashcard 23: Find the derivative of $f(x) = \text{sin }(2x)$.

Answer: $f'(x) = 2\text{cos }(2x)$. Use chain rule with inner function $2x$.

Flashcard 24: Find the second derivative of $y = \text{cos } x$.

Answer: $\frac{d^2y}{dx^2} = -\text{cos } x$. Differentiate $f'(x) = -\text{sin } x$ to get $f''(x) = -\text{cos } x$.

Flashcard 25: Differentiate $f(x) = \text{ln } x$ at $x = e$.

Answer: $f'(e) = \frac{1}{e}$. $f'(x) = \frac{1}{x}$, so $f'(e) = \frac{1}{e}$.

Flashcard 26: Find the derivative of $f(x) = \text{ln }(5x)$.

Answer: $f'(x) = \frac{1}{x}$. Constant multiple rule: $\frac{d}{dx}[\text{ln}(5x)] = \frac{1}{5x} \cdot 5 = \frac{1}{x}$.

Flashcard 27: What is the derivative of $e^x$?

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

Flashcard 28: Differentiate $f(x) = 5e^x + 7\text{ln } x$.

Answer: $f'(x) = 5e^x + \frac{7}{x}$. Use sum rule with constant multiples.

Flashcard 29: Determine the derivative of $f(x) = x\text{cos } x$.

Answer: $f'(x) = \text{cos } x - x\text{sin } x$. Use product rule: $(uv)' = u'v + uv'$.

Flashcard 30: Find the derivative: $y = e^x$.

Answer: $\frac{dy}{dx} = e^x$. Apply the derivative rule for $e^x$.

Flashcard 31: Evaluate $f'(x)$ for $f(x) = e^x - \text{ln } x$ at $x = 1$.

Answer: $f'(1) = e - 1$. $f'(x) = e^x - \frac{1}{x}$, so $f'(1) = e - 1$.

Flashcard 32: Determine $f'(x)$ if $f(x) = \cos x + \sin x$.

Answer: $f'(x) = -\sin x + \cos x$. Use sum rule: derivative of each term separately.

Flashcard 33: Evaluate the derivative: $f(x) = \text{ln } x$ at $x = 2$.

Answer: $f'(2) = \frac{1}{2}$. $f'(x) = \frac{1}{x}$, so $f'(2) = \frac{1}{2}$.

Flashcard 34: Determine $f'(x)$ if $f(x) = e^x + \text{ln } x$.

Answer: $f'(x) = e^x + \frac{1}{x}$. Use sum rule: derivative of each term separately.

Flashcard 35: Differentiate $f(x) = \text{cos } x$ at $x = 0$.

Answer: $f'(0) = 0$. $f'(x) = -\text{sin } x$, so $f'(0) = -\text{sin }(0) = 0$.

Flashcard 36: Find the slope of the tangent to $y = e^x$ at $x = 2$.

Answer: Slope = $e^2$. The slope equals the derivative at the point.

Flashcard 37: Evaluate the derivative: $f(x) = e^x$ at $x = 0$.

Answer: $f'(0) = 1$. $f'(x) = e^x$, so $f'(0) = e^0 = 1$.

Flashcard 38: What is the derivative of $\text{ln } x$?

Answer: $\frac{1}{x}$. Standard derivative of natural logarithm.

Flashcard 39: Evaluate the derivative: $f(x) = e^x$ at $x = 0$.

Answer: $f'(0) = 1$. $f'(x) = e^x$, so $f'(0) = e^0 = 1$.

Flashcard 40: What is the derivative of $\text{ln } x$?

Answer: $\frac{1}{x}$. Standard derivative of natural logarithm.

Flashcard 41: Determine the derivative of $f(x) = x\text{ln } x$.

Answer: $f'(x) = \text{ln } x + 1$. Use product rule: $(uv)' = u'v + uv'$.

Flashcard 42: Determine the derivative of $f(x) = x\text{cos } x$.

Answer: $f'(x) = \text{cos } x - x\text{sin } x$. Use product rule: $(uv)' = u'v + uv'$.

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

Answer: $f'(x) = \text{sin } x + x\text{cos } x$. Use product rule: $(uv)' = u'v + uv'$.

Flashcard 44: Differentiate $f(x) = 3\text{cos } x - 2\text{sin } x$.

Answer: $f'(x) = -3\text{sin } x - 2\text{cos } x$. Use sum/difference rule with constant multiples.

Flashcard 45: Evaluate $f'(x)$ for $f(x) = e^x - \text{ln } x$ at $x = 1$.

Answer: $f'(1) = e - 1$. $f'(x) = e^x - \frac{1}{x}$, so $f'(1) = e - 1$.

Flashcard 46: Find the derivative: $y = e^x$.

Answer: $\frac{dy}{dx} = e^x$. Apply the derivative rule for $e^x$.

Flashcard 47: Evaluate the derivative: $f(x) = e^x$ at $x = \text{ln } 2$.

Answer: $f'(\text{ln } 2) = 2$. $f'(x) = e^x$, so $f'(\text{ln } 2) = e^{\text{ln } 2} = 2$.

Flashcard 48: Differentiate $f(x) = 3\text{cos } x - 2\text{sin } x$.

Answer: $f'(x) = -3\text{sin } x - 2\text{cos } x$. Use sum/difference rule with constant multiples.

Flashcard 49: Find the slope of the tangent to $y = \text{ln } x$ at $x = 4$.

Answer: Slope = $\frac{1}{4}$. The slope equals the derivative at the point.

Flashcard 50: Find the second derivative of $y = e^x$.

Answer: $\frac{d^2y}{dx^2} = e^x$. Differentiate $f'(x) = e^x$ to get $f''(x) = e^x$.

Flashcard 51: What is the derivative of $\text{cos } x$?

Answer: $-\text{sin } x$. Basic derivative rule for cosine function.

Flashcard 52: Differentiate $f(x) = \text{cos } x$ at $x = 0$.

Answer: $f'(0) = 0$. $f'(x) = -\text{sin } x$, so $f'(0) = -\text{sin }(0) = 0$.

Flashcard 53: Differentiate $f(x) = 5e^x + 7\text{ln } x$.

Answer: $f'(x) = 5e^x + \frac{7}{x}$. Use sum rule with constant multiples.

Flashcard 54: Find the derivative of $f(x) = \sin(2x)$.

Answer: $f'(x) = 2\cos(2x)$. Use chain rule with inner function $2x$.

Flashcard 55: Evaluate the derivative: $f(x) = \text{sin } x$ at $x = 0$.

Answer: $f'(0) = 1$. $f'(x) = \text{cos } x$, so $f'(0) = \text{cos}(0) = 1$.

Flashcard 56: What is the derivative of $e^x$?

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

Flashcard 57: Determine the derivative of $f(x) = x\text{ln } x$.

Answer: $f'(x) = \text{ln } x + 1$. Use product rule: $(uv)' = u'v + uv'$.

Flashcard 58: Evaluate $f'(x)$ for $f(x) = \text{cos } x - e^x$ at $x = 0$.

Answer: $f'(0) = -1$. $f'(x) = -\text{sin } x - e^x$, so $f'(0) = 0 - 1 = -1$.

Flashcard 59: Determine the derivative of $f(x) = xe^x$.

Answer: $f'(x) = e^x + xe^x$. Use product rule: $(uv)' = u'v + uv'$.

Flashcard 60: Find the second derivative of $y = \text{cos } x$.

Answer: $\frac{d^2y}{dx^2} = -\text{cos } x$. Differentiate $f'(x) = -\text{sin } x$ to get $f''(x) = -\text{cos } x$.

Flashcard 61: Find the second derivative of $y = \text{sin } x$.

Answer: $\frac{d^2y}{dx^2} = -\text{sin } x$. Differentiate $f'(x) = \text{cos } x$ to get $f''(x) = -\text{sin } x$.

Flashcard 62: What is the derivative of $\text{cos } x$?

Answer: $-\text{sin } x$. Basic derivative rule for cosine function.

Flashcard 63: Evaluate the derivative: $f(x) = \text{ln } x$ at $x = 1$.

Answer: $f'(1) = 1$. $f'(x) = \frac{1}{x}$, so $f'(1) = \frac{1}{1} = 1$.

Flashcard 64: Find the second derivative of $y = \text{ln } x$.

Answer: $\frac{d^2y}{dx^2} = -\frac{1}{x^2}$. Differentiate $f'(x) = \frac{1}{x}$ to get $f''(x) = -\frac{1}{x^2}$.

Flashcard 65: Find the derivative: $y = \text{sin } x$.

Answer: $\frac{dy}{dx} = \text{cos } x$. Apply the derivative rule for $\text{sin } x$.

Flashcard 66: Determine $f'(x)$ if $f(x) = e^x + \text{ln } x$.

Answer: $f'(x) = e^x + \frac{1}{x}$. Use sum rule: derivative of each term separately.

Flashcard 67: What is the derivative of $\text{sin } x$?

Answer: $\text{cos } x$. Basic derivative rule for sine function.

Flashcard 68: Find the derivative: $y = \text{sin } x$.

Answer: $\frac{dy}{dx} = \text{cos } x$. Apply the derivative rule for $\text{sin } x$.

Flashcard 69: Differentiate $f(x) = e^x$ at $x = 1$.

Answer: $f'(1) = e$. $f'(x) = e^x$, so $f'(1) = e^1 = e$.

Flashcard 70: Differentiate $f(x) = \text{ln } x$ at $x = e$.

Answer: $f'(e) = \frac{1}{e}$. $f'(x) = \frac{1}{x}$, so $f'(e) = \frac{1}{e}$.

Flashcard 71: Find the derivative of $f(x) = \text{cos }(3x)$.

Answer: $f'(x) = -3\text{sin }(3x)$. Use chain rule with inner function $3x$.

Flashcard 72: Find the slope of the tangent to $y = e^x$ at $x = 2$.

Answer: Slope = $e^2$. The slope equals the derivative at the point.

Flashcard 73: Evaluate the derivative: $f(x) = \text{cos } x$ at $x = \frac{\text{π}}{2}$.

Answer: $f'\big(\frac{\text{π}}{2}\big) = -1$. $f'(x) = -\text{sin } x$, so $f'\big(\frac{\text{π}}{2}\big) = -\text{sin}\big(\frac{\text{π}}{2}\big) = -1$.

Flashcard 74: Find the derivative: $y = \text{cos } x$.

Answer: $\frac{dy}{dx} = -\text{sin } x$. Apply the derivative rule for $\text{cos } x$.

Flashcard 75: Determine the derivative of $f(x) = x\text{sin } x$.

Answer: $f'(x) = \text{sin } x + x\text{cos } x$. Use product rule: $(uv)' = u'v + uv'$.

Flashcard 76: Evaluate the derivative: $f(x) = e^x$ at $x = \text{ln } 2$.

Answer: $f'(\text{ln } 2) = 2$. $f'(x) = e^x$, so $f'(\text{ln } 2) = e^{\text{ln } 2} = 2$.