AP Calculus BC Flashcards: Derivative Notation

What this deck covers

This deck focuses on Derivative Notation, 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: Find the derivative of $f(x) = \sqrt{x^3}$.

Answer: $f'(x) = \frac{3}{2}x^{1/2}$. Rewrite $\sqrt{x^3} = x^{3/2}$ and use power rule.

Flashcard 2: How is the derivative of $f(x)$ denoted using Newton's notation?

Answer: $\dot{f}(x)$. Dot notation represents time derivative in physics.

Flashcard 3: Find the derivative of $f(x) = \ln(\sin x)$.

Answer: $f'(x) = \cot x$. Chain rule with $\ln(\sin x)$.

Flashcard 4: Find the derivative of $f(x) = 3x^2 + 2x$.

Answer: $f'(x) = 6x + 2$. Apply power rule to each term separately.

Flashcard 5: Find the derivative of $f(x) = \frac{1}{3}x^3$.

Answer: $f'(x) = x^2$. Constant factor $\frac{1}{3}$ remains, apply power rule.

Flashcard 6: How is the derivative of $f(x)$ at $x = a$ denoted using prime notation?

Answer: $f'(a)$. Prime notation denotes derivative at specific point.

Flashcard 7: What is the derivative notation using Leibniz's notation for $y = f(x)$?

Answer: $\frac{dy}{dx}$. Leibniz notation shows derivative of $y$ with respect to $x$.

Flashcard 8: Find the derivative of $f(x) = 2\sec x$.

Answer: $f'(x) = 2\sec x \tan x$. Constant multiple of secant derivative.

Flashcard 9: Find the derivative of $f(x) = 3x^2 + 2x$.

Answer: $f'(x) = 6x + 2$. Apply power rule to each term separately.

Flashcard 10: Find the derivative of $f(x) = 5e^x - 4$.

Answer: $f'(x) = 5e^x$. Constant multiple rule with exponential derivative.

Flashcard 11: Find the derivative of $f(x) = x \cos x$.

Answer: $f'(x) = \cos x - x \sin x$. Product rule with $u = x$ and $v = \cos x$.

Flashcard 12: State the formula for the derivative of $f(x) = x^n$.

Answer: $f'(x) = nx^{n-1}$. Power rule: bring down exponent, reduce power by 1.

Flashcard 13: State the formula for the derivative of $f(x) = x^n$.

Answer: $f'(x) = nx^{n-1}$. Power rule: bring down exponent, reduce power by 1.

Flashcard 14: What is the derivative of $f(x) = \cot x$?

Answer: $-\csc^2 x$. Derivative of cotangent is negative cosecant squared.

Flashcard 15: Find the derivative of $f(x) = 8x^{-1/2}$.

Answer: $f'(x) = -4x^{-3/2}$. Constant 8 times power rule on $x^{-1/2}$.

Flashcard 16: How is the derivative of $f(x)$ at $x = a$ denoted using prime notation?

Answer: $f'(a)$. Prime notation denotes derivative at specific point.

Flashcard 17: Find the derivative of $f(x) = x^2 \sin x$.

Answer: $2x \sin x + x^2 \cos x$. Apply product rule: $(uv)' = u'v + uv'$.

Flashcard 18: Find the derivative of $f(x) = x^4 - 3x^2 + x$.

Answer: $f'(x) = 4x^3 - 6x + 1$. Apply power rule to each term.

Flashcard 19: Find the derivative of $f(x) = x^4 - 3x^2 + x$.

Answer: $f'(x) = 4x^3 - 6x + 1$. Apply power rule to each term.

Flashcard 20: Find the derivative of $f(x) = x^3 - 5x + 4$.

Answer: $f'(x) = 3x^2 - 5$. Derivative of constant is 0, apply power rule to other terms.

Flashcard 21: Find the derivative of $f(x) = x^3 - 5x + 4$.

Answer: $f'(x) = 3x^2 - 5$. Derivative of constant is 0, apply power rule to other terms.

Flashcard 22: What does the derivative represent geometrically?

Answer: Slope of the tangent line. Derivative gives instantaneous rate of change.

Flashcard 23: What is the derivative of $f(x) = \cot x$?

Answer: $- \csc^2 x$. Derivative of cotangent is negative cosecant squared.

Flashcard 24: Find the derivative of $f(x) = 5e^x - 4$.

Answer: $f'(x) = 5e^x$. Constant multiple rule with exponential derivative.

Flashcard 25: What is the derivative of $f(x) = a^x$ where $a > 0$?

Answer: $a^x \ln a$. Exponential with base $a$ requires $\ln a$ factor.

Flashcard 26: What is the derivative of a constant function $f(x) = c$?

Answer:

1. Constants have zero rate of change.

Flashcard 27: Find the derivative of $f(x) = 7x^5$.

Answer: $f'(x) = 35x^4$. Constant multiple of power rule.

Flashcard 28: What is the derivative of $f(x) = e^x$?

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

Flashcard 29: Find the derivative of $f(x) = \frac{1}{x}$

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

Flashcard 30: Find the derivative of $f(x) = \ln(\sin x)$.

Answer: $f'(x) = \cot x$. Chain rule with $\ln(\sin x)$.

Flashcard 31: Find the derivative of $f(x) = \frac{1}{2}x^{-2}$.

Answer: $f'(x) = -x^{-3}$. Constant $\frac{1}{2}$ times power rule on $x^{-2}$.

Flashcard 32: Find the derivative of $f(x) = \frac{x^2 + 1}{x}$.

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

Flashcard 33: How is the derivative of $f(x)$ denoted using Newton's notation?

Answer: $\\dot{f}(x)$. Dot notation represents time derivative in physics.

Flashcard 34: Find the derivative of $f(x) = \sqrt{x}$.

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

Flashcard 35: What is the derivative of $f(x) = \sin x$?

Answer: $\cos x$. Derivative of sine is cosine.

Flashcard 36: Find the derivative of $f(x) = \ln(x^2 + 1)$.

Answer: $\frac{2x}{x^2 + 1}$. Use chain rule with $\ln$ and $x^2 + 1$.

Flashcard 37: Find 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 38: Find the derivative of $f(x) = 2\sec x$.

Answer: $f'(x) = 2\sec x \tan x$. Constant multiple of secant derivative.

Flashcard 39: What is the derivative of $f(x) = \cos x$?

Answer: $-\sin x$. Derivative of cosine is negative sine.

Flashcard 40: Find the derivative of $f(x) = 7x^5$.

Answer: $f'(x) = 35x^4$. Constant multiple of power rule.

Flashcard 41: Find the derivative of $f(x) = x \cos x$.

Answer: $f'(x) = \cos x - x \sin x$. Product rule with $u = x$ and $v = \cos x$.

Flashcard 42: What is the derivative of $f(x) = \csc x$?

Answer: $- \csc x \cot x$. Derivative is negative product of cosecant and cotangent.

Flashcard 43: Find the derivative of $f(x) = \ln(x^2 + 1)$.

Answer: $\frac{2x}{x^2 + 1}$. Use chain rule with $\ln$ and $x^2 + 1$.

Flashcard 44: Find the derivative of $f(x) = \frac{1}{x}$.

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

Flashcard 45: Find the derivative of $f(x) = \sqrt{x}$.

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

Flashcard 46: What is the derivative of $f(x) = \ln x$?

Answer: $\frac{1}{x}$. Natural log derivative is reciprocal function.

Flashcard 47: Find the derivative of $f(x) = \frac{x^2 + 1}{x}$.

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

Flashcard 48: Find the derivative of $f(x) = \sin^2 x$.

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

Flashcard 49: What is the derivative of $f(x) = \sec x$?

Answer: $\sec x \tan x$. Derivative involves product of secant and tangent.

Flashcard 50: What is the limit definition of the derivative of $f(x)$ at $x = a$?

Answer: $f'(a) = \lim_{{h \to 0}} \frac{f(a+h) - f(a)}{h}$. Limit of difference quotient as $h$ approaches 0.

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

Answer: $-\sin x$. Derivative of cosine is negative sine.

Flashcard 52: Find 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 53: What is the derivative of $f(x) = \csc x$?

Answer: $-\csc x \cot x$. Derivative is negative product of cosecant and cotangent.

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

Answer: $\sec^2 x$. Derivative of tangent is secant squared.

Flashcard 55: What is the derivative of $f(x) = \sec x$?

Answer: $\sec x \tan x$. Derivative involves product of secant and tangent.

Flashcard 56: Find the derivative of $f(x) = \tan(x^2)$.

Answer: $f'(x) = 2x \sec^2(x^2)$. Chain rule: outer derivative times inner derivative.

Flashcard 57: What is the derivative of $f(x) = \ln x$?

Answer: $\frac{1}{x}$. Natural log derivative is reciprocal function.

Flashcard 58: Find the derivative of $f(x) = \frac{1}{3}x^3$.

Answer: $f'(x) = x^2$. Constant factor $\frac{1}{3}$ remains, apply power rule.

Flashcard 59: What is the derivative of $f(x) = \sin x$?

Answer: $\cos x$. Derivative of sine is cosine.

Flashcard 60: What is the limit definition of the derivative of $f(x)$ at $x = a$?

Answer: $f'(a) = \lim_{{h \to 0}} \frac{f(a+h) - f(a)}{h}$. Limit of difference quotient as $h$ approaches 0.

Flashcard 61: Find the derivative of $f(x) = \sqrt{x^3}$.

Answer: $f'(x) = \frac{3}{2}x^{1/2}$. Rewrite $\sqrt{x^3} = x^{3/2}$ and use power rule.

Flashcard 62: Find the derivative of $f(x) = \sin^2 x$.

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

Flashcard 63: Find the derivative of $f(x) = \tan(x^2)$.

Answer: $f'(x) = 2x \sec^2(x^2)$. Chain rule: outer derivative times inner derivative.

Flashcard 64: What is the derivative of $f(x) = a^x$ where $a > 0$?

Answer: $a^x \ln a$. Exponential with base $a$ requires $\ln a$ factor.

Flashcard 65: Find the derivative of $f(x) = x^2 \sin x$.

Answer: $2x \sin x + x^2 \cos x$. Apply product rule: $(uv)' = u'v + uv'$.

Flashcard 66: What is the derivative of $f(x) = \tan x$?

Answer: $\sec^2 x$. Derivative of tangent is secant squared.

Flashcard 67: Find the derivative of $f(x) = 8x^{-1/2}$.

Answer: $f'(x) = -4x^{-3/2}$. Constant 8 times power rule on $x^{-1/2}$.

Flashcard 68: What is the derivative notation using Leibniz's notation for $y = f(x)$?

Answer: $\frac{dy}{dx}$. Leibniz notation shows derivative of $y$ with respect to $x$.