AP Calculus AB Flashcards: Derivatives Of Reciprocal Trig Functions

What this deck covers

This deck focuses on Derivatives Of Reciprocal Trig Functions, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

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 $\tan(x)$?

Answer: $\sec^2(x)$. Standard derivative formula for tangent function.

Flashcard 2: Find the derivative of $h(x) = \csc(\ln(x))$.

Answer: $-\frac{1}{x}\csc(\ln(x))\cot(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.

Flashcard 3: Determine the derivative of $y = \tan(3x^2)$.

Answer: $6x\sec^2(3x^2)$. Chain rule: multiply by derivative of inner function $6x$.

Flashcard 4: Find $d/dx$ for $f(x) = \sec^2(x)$.

Answer: $2\sec(x)\sec(x)\tan(x)$. Chain rule applied to $\sec^2(x) = 2\sec(x) \cdot \sec(x)\tan(x)$.

Flashcard 5: Find the derivative of $f(x) = \tan(x^2)$.

Answer: $2x\sec^2(x^2)$. Chain rule: multiply by derivative of inner function $2x$.

Flashcard 6: Determine $d/dx$ for $y = \csc(e^x)$.

Answer: $-e^x\csc(e^x)\cot(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.

Flashcard 7: Determine $d/dx$ for $h(x) = \csc(2x)$.

Answer: $-2\csc(2x)\cot(2x)$. Chain rule: multiply by derivative of inner function $2$.

Flashcard 8: Find $d/dx$ for $f(x) = \sec(\sin(x))$.

Answer: $\cos(x)\sec(\sin(x))\tan(\sin(x))$. Chain rule: multiply by derivative of $\sin(x)$ which is $\cos(x)$.

Flashcard 9: What is the derivative of $f(x) = \tan(\frac{1}{x})$?

Answer: $-\frac{1}{x^2}\sec^2(\frac{1}{x})$. Chain rule: multiply by derivative of $\frac{1}{x}$ which is $-\frac{1}{x^2}$.

Flashcard 10: What is the derivative of $f(x) = \cot(2x + 3)$?

Answer: $-2\csc^2(2x + 3)$. Chain rule: multiply by derivative of inner function $2$.

Flashcard 11: Determine the derivative of $y = \csc(4x^2)$.

Answer: $-8x\csc(4x^2)\cot(4x^2)$. Chain rule: multiply by derivative of inner function $8x$.

Flashcard 12: Calculate $d/dx$ for $g(x) = \sec(x^5)$.

Answer: $5x^4\sec(x^5)\tan(x^5)$. Chain rule: multiply by derivative of inner function $5x^4$.

Flashcard 13: Find $d/dx$ for $g(x) = \cot(\ln(x))$.

Answer: $-\frac{1}{x}\csc^2(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.

Flashcard 14: Calculate $d/dx$ for $g(x) = \sec(x^5)$.

Answer: $5x^4\sec(x^5)\tan(x^5)$. Chain rule: multiply by derivative of inner function $5x^4$.

Flashcard 15: Compute the derivative of $g(x) = \sec(5x)$.

Answer: $5\sec(5x)\tan(5x)$. Chain rule: multiply by derivative of inner function $5$.

Flashcard 16: State the derivative of $\cot(x)$.

Answer: $-\csc^2(x)$. Standard derivative formula for cotangent function.

Flashcard 17: What is the derivative of $\tan(x)$?

Answer: $\sec^2(x)$. Standard derivative formula for tangent function.

Flashcard 18: State the derivative of $g(x) = \cot(2x^3)$.

Answer: $-6x^2\csc^2(2x^3)$. Chain rule: multiply by derivative of inner function $6x^2$.

Flashcard 19: Identify the derivative of $f(x) = \tan(2x + 1)$.

Answer: $2\sec^2(2x + 1)$. Chain rule: multiply by derivative of inner function $2$.

Flashcard 20: State the derivative of $g(x) = \sec(x^3)$.

Answer: $3x^2\sec(x^3)\tan(x^3)$. Chain rule: multiply by derivative of inner function $3x^2$.

Flashcard 21: State the derivative of $g(x) = \sec(x^3)$.

Answer: $3x^2\sec(x^3)\tan(x^3)$. Chain rule: multiply by derivative of inner function $3x^2$.

Flashcard 22: Compute the derivative of $h(x) = \sec(\ln(x))$.

Answer: $\frac{1}{x}\sec(\ln(x))\tan(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.

Flashcard 23: Calculate the derivative of $y = \csc(4x)$.

Answer: $-4\csc(4x)\cot(4x)$. Chain rule: multiply by derivative of inner function $4$.

Flashcard 24: Determine the derivative of $y = \csc(4x^2)$.

Answer: $-8x\csc(4x^2)\cot(4x^2)$. Chain rule: multiply by derivative of inner function $8x$.

Flashcard 25: State $d/dx$ for $g(x) = \cot(\sqrt{x})$.

Answer: $-\frac{1}{2\sqrt{x}}\csc^2(\sqrt{x})$. Chain rule: multiply by derivative of $\sqrt{x}$ which is $\frac{1}{2\sqrt{x}}$.

Flashcard 26: What is the derivative of $f(x) = \tan(x^2 + 1)$?

Answer: $2x\sec^2(x^2 + 1)$. Chain rule: multiply by derivative of inner function $2x$.

Flashcard 27: State the derivative of $\cot(x)$.

Answer: $-\csc^2(x)$. Standard derivative formula for cotangent function.

Flashcard 28: What is the derivative of $y = \tan(\sqrt{x})$?

Answer: $\frac{1}{2\sqrt{x}}\sec^2(\sqrt{x})$. Chain rule: multiply by derivative of $\sqrt{x}$ which is $\frac{1}{2\sqrt{x}}$.

Flashcard 29: Find the derivative of $f(x) = \tan(x^2)$.

Answer: $2x\sec^2(x^2)$. Chain rule: multiply by derivative of inner function $2x$.

Flashcard 30: Determine the derivative of $y = \tan(3x^2)$.

Answer: $6x\sec^2(3x^2)$. Chain rule: multiply by derivative of inner function $6x$.

Flashcard 31: Calculate the derivative of $y = \csc(4x)$.

Answer: $-4\csc(4x)\cot(4x)$. Chain rule: multiply by derivative of inner function $4$.

Flashcard 32: What is the derivative of $y = \tan(\sqrt{x})$?

Answer: $\frac{1}{2\sqrt{x}}\sec^2(\sqrt{x})$. Chain rule: multiply by derivative of $\sqrt{x}$ which is $\frac{1}{2\sqrt{x}}$.

Flashcard 33: What is the derivative of $y = \cot(e^x)$?

Answer: $-e^x\csc^2(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.

Flashcard 34: State the derivative of $f(x) = \cot(3x^3)$.

Answer: $-9x^2\csc^2(3x^3)$. Chain rule: multiply by derivative of inner function $9x^2$.

Flashcard 35: State $d/dx$ for $g(x) = \cot(\sqrt{x})$.

Answer: $-\frac{1}{2\sqrt{x}}\csc^2(\sqrt{x})$. Chain rule: multiply by derivative of $\sqrt{x}$ which is $\frac{1}{2\sqrt{x}}$.

Flashcard 36: Compute $d/dx$ for $g(x) = \sec(x^2 + 1)$.

Answer: $2x\sec(x^2 + 1)\tan(x^2 + 1)$. Chain rule: multiply by derivative of inner function $2x$.

Flashcard 37: Identify the derivative of $f(x) = \tan(2x + 1)$.

Answer: $2\sec^2(2x + 1)$. Chain rule: multiply by derivative of inner function $2$.

Flashcard 38: What is the derivative of $f(x) = \tan(x^2 + 1)$?

Answer: $2x\sec^2(x^2 + 1)$. Chain rule: multiply by derivative of inner function $2x$.

Flashcard 39: What is the derivative of $f(x) = \cot(3x)$?

Answer: $-3\csc^2(3x)$. Chain rule: multiply by derivative of inner function $3$.

Flashcard 40: State the derivative of $f(x) = \cot(3x^3)$.

Answer: $-9x^2\csc^2(3x^3)$. Chain rule: multiply by derivative of inner function $9x^2$.

Flashcard 41: What is the derivative of $f(x) = \cot(2x + 3)$?

Answer: $-2\csc^2(2x + 3)$. Chain rule: multiply by derivative of inner function $2$.

Flashcard 42: Determine $d/dx$ for $h(x) = \csc(2x)$.

Answer: $-2\csc(2x)\cot(2x)$. Chain rule: multiply by derivative of inner function $2$.

Flashcard 43: Find the derivative of $f(x) = \tan(e^x)$.

Answer: $e^x\sec^2(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.

Flashcard 44: Find the derivative of $f(x) = \tan(e^x)$.

Answer: $e^x\sec^2(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.

Flashcard 45: Identify the derivative of $y = \tan(x + \pi)$.

Answer: $\sec^2(x + \pi)$. Chain rule: multiply by derivative of inner function $1$.

Flashcard 46: Determine the derivative of $y = \csc(x^2)$.

Answer: $-2x\csc(x^2)\cot(x^2)$. Chain rule: multiply by derivative of inner function $2x$.

Flashcard 47: What is the derivative of $f(x) = \tan(\frac{1}{x})$?

Answer: $-\frac{1}{x^2}\sec^2(\frac{1}{x})$. Chain rule: multiply by derivative of $\frac{1}{x}$ which is $-\frac{1}{x^2}$

Flashcard 48: Find $d/dx$ for $g(x) = \cot(\ln(x))$.

Answer: $-\frac{1}{x}\csc^2(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.

Flashcard 49: Calculate the derivative of $h(x) = \sec(\sqrt{x})$.

Answer: $\frac{1}{2\sqrt{x}}\sec(\sqrt{x})\tan(\sqrt{x})$. Chain rule: multiply by derivative of $\sqrt{x}$ which is $\frac{1}{2\sqrt{x}}$.

Flashcard 50: What is the derivative of $y = \cot(x^3)$?

Answer: $-3x^2\csc^2(x^3)$. Chain rule: multiply by derivative of inner function $3x^2$.

Flashcard 51: Find $d/dx$ for $f(x) = \sec(\sin(x))$.

Answer: $\cos(x)\sec(\sin(x))\tan(\sin(x))$. Chain rule: multiply by derivative of $\sin(x)$ which is $\cos(x)$.

Flashcard 52: What is the derivative of $f(x) = \cot(3x)$?

Answer: $-3\csc^2(3x)$. Chain rule: multiply by derivative of inner function $3$.

Flashcard 53: What is the derivative of $y = \cot(e^x)$?

Answer: $-e^x\csc^2(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.

Flashcard 54: What is the derivative of $y = \cot(x^3)$?

Answer: $-3x^2\csc^2(x^3)$. Chain rule: multiply by derivative of inner function $3x^2$.

Flashcard 55: Find $d/dx$ for $h(x) = \sec(3x + 1)$.

Answer: $3\sec(3x + 1)\tan(3x + 1)$. Chain rule: multiply by derivative of inner function $3$.

Flashcard 56: Compute the derivative of $g(x) = \csc(x^4)$.

Answer: $-4x^3\csc(x^4)\cot(x^4)$. Chain rule: multiply by derivative of inner function $4x^3$.

Flashcard 57: Determine the derivative of $y = \csc(x^2)$.

Answer: $-2x\csc(x^2)\cot(x^2)$. Chain rule: multiply by derivative of inner function $2x$.

Flashcard 58: State the derivative of $g(x) = \cot(2x^3)$.

Answer: $-6x^2\csc^2(2x^3)$. Chain rule: multiply by derivative of inner function $6x^2$.

Flashcard 59: What is the derivative of $f(x) = \tan(5x^2)$?

Answer: $10x\sec^2(5x^2)$. Chain rule: multiply by derivative of inner function $10x$.

Flashcard 60: Compute the derivative of $h(x) = \sec(\ln(x))$.

Answer: $\frac{1}{x}\sec(\ln(x))\tan(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.

Flashcard 61: What is the derivative of $\sec(x)$?

Answer: $\sec(x)\tan(x)$. Standard derivative formula for secant function.

Flashcard 62: Compute $d/dx$ for $g(x) = \sec(x^2 + 1)$.

Answer: $2x\sec(x^2 + 1)\tan(x^2 + 1)$. Chain rule: multiply by derivative of inner function $2x$.

Flashcard 63: What is the derivative of $\sec(x)$?

Answer: $\sec(x)\tan(x)$. Standard derivative formula for secant function.

Flashcard 64: What is the derivative of $f(x) = \tan(5x^2)$?

Answer: $10x\sec^2(5x^2)$. Chain rule: multiply by derivative of inner function $10x$.

Flashcard 65: Determine $d/dx$ for $y = \csc(e^x)$.

Answer: $-e^x\csc(e^x)\cot(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.

Flashcard 66: Calculate the derivative of $h(x) = \sec(\sqrt{x})$.

Answer: $\frac{1}{2\sqrt{x}}\sec(\sqrt{x})\tan(\sqrt{x})$. Chain rule: multiply by derivative of $\sqrt{x}$ which is $\frac{1}{2\sqrt{x}}$.

Flashcard 67: Identify the derivative of $y = \tan(x + \pi)$.

Answer: $\sec^2(x + \pi)$. Chain rule: multiply by derivative of inner function $1$.

Flashcard 68: Compute the derivative of $g(x) = \csc(x^4)$.

Answer: $-4x^3\csc(x^4)\cot(x^4)$. Chain rule: multiply by derivative of inner function $4x^3$.

Flashcard 69: State the derivative of $\csc(x)$.

Answer: $-\csc(x)\cot(x)$. Standard derivative formula for cosecant function.

Flashcard 70: Find $d/dx$ for $h(x) = \sec(3x + 1)$.

Answer: $3\sec(3x + 1)\tan(3x + 1)$. Chain rule: multiply by derivative of inner function $3$.

Flashcard 71: Find the derivative of $h(x) = \csc(\ln(x))$.

Answer: $-\frac{1}{x}\csc(\ln(x))\cot(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.

Flashcard 72: State the derivative of $\csc(x)$.

Answer: $-\csc(x)\cot(x)$. Standard derivative formula for cosecant function.

Flashcard 73: Compute the derivative of $g(x) = \sec(5x)$.

Answer: $5\sec(5x)\tan(5x)$. Chain rule: multiply by derivative of inner function $5$.

Flashcard 74: Find $d/dx$ for $f(x) = \sec^2(x)$.

Answer: $2\sec(x)\sec(x)\tan(x)$. Chain rule applied to $\sec^2(x) = 2\sec(x) \cdot \sec(x)\tan(x)$.