AP Calculus AB Flashcards: Differentiating Inverse Trigonometric Functions

What this deck covers

This deck focuses on Differentiating Inverse Trigonometric 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: State the derivative of $\text{arccsc}(x)$.

Answer: $-\frac{1}{|x| \sqrt{x^2 - 1}}$ Standard derivative formula for inverse cosecant function.

Flashcard 2: Determine the derivative of $y = \text{arccot}(\text{cot}(x))$.

Answer: $-1$. Inverse function cancels on principal domain.

Flashcard 3: Evaluate the derivative of $\text{arccot}(x)$ at $x = 1$.

Answer: $-\frac{1}{2}$. Substitute $x = 1$ into $-\frac{1}{1+x^2}$ formula.

Flashcard 4: Find the derivative of $y = \text{arccot}(\frac{1}{x})$ at $x = 1$.

Answer: $-1$. Use chain rule with $\frac{d}{dx}[\frac{1}{x}] = -\frac{1}{x^2}$ and evaluate at $x = 1$.

Flashcard 5: Determine the derivative of $y = \text{arcsin}(\text{sin}(x))$.

Answer: $1$. Inverse function cancels on principal domain.

Flashcard 6: What is the derivative of $\text{arctan}(x)$ at $x = 1$?

Answer: $\frac{1}{2}$. Substitute $x = 1$ into $\frac{1}{1+x^2}$ formula.

Flashcard 7: Determine the derivative of $y = \text{arccsc}(\text{csc}(x))$.

Answer: $-1$. Inverse function cancels on principal domain.

Flashcard 8: State the derivative of $\text{arcsin}(x)$.

Answer: $\frac{1}{\text{sqrt}(1-x^2)}$. Standard derivative formula for inverse sine function.

Flashcard 9: State the derivative of $\text{arcsin}(x)$.

Answer: \frac{1}{\sqrt{1 - x^2)}. Standard derivative formula for inverse sine function.

Flashcard 10: Evaluate the derivative of $y = \text{arccos}(\frac{x}{3})$ at $x = 1$.

Answer: $-\frac{1}{\sqrt{8}}$. Use chain rule with $\frac{d}{dx}[\frac{x}{3}] = \frac{1}{3}$ and evaluate at $x = 1$.

Flashcard 11: Determine the derivative of $y = \text{arcsec}(\text{sec}(x))$.

Answer: $1$. Inverse function cancels on principal domain.

Flashcard 12: Identify the derivative of $\text{arcsec}(6x)$ with respect to $x$.

Answer: $\frac{6}{|6x| \sqrt{36x^2 - 1}}$. Apply chain rule with arcsec derivative formula.

Flashcard 13: State the derivative of $\text{arctan}(x)$.

Answer: $\frac{1}{1+x^2}$. Standard derivative formula for inverse tangent function.

Flashcard 14: Evaluate the derivative of $\text{arccsc}(x)$ at $x = -2$.

Answer: $\frac{1}{2\text{sqrt}(3)}$. Substitute $x = -2$ into $-\frac{1}{|x|\sqrt{x^2-1}}$ formula.

Flashcard 15: Identify the derivative of $\text{arctan}(5x)$ with respect to $x$.

Answer: $\frac{5}{1+25x^2}$. Apply chain rule: derivative of $5x$ times $\frac{1}{1+(5x)^2}$.

Flashcard 16: Evaluate the derivative of $y = \text{arccos}(\frac{x}{3})$ at $x = 1$.

Answer: $-\frac{1}{\sqrt{8}}$. Use chain rule with $\frac{d}{dx}[\frac{x}{3}] = \frac{1}{3}$ and evaluate at $x = 1$.

Flashcard 17: Evaluate the derivative of $y = \text{arctan}(2x)$ at $x = 0.5$.

Answer: $\frac{2}{5}$. Use chain rule with $\frac{d}{dx}[2x] = 2$ and evaluate at $x = 0.5$.

Flashcard 18: Evaluate the derivative of $\text{arccot}(x)$ at $x = 1$.

Answer: $-\frac{1}{2}$. Substitute $x = 1$ into $-\frac{1}{1+x^2}$ formula.

Flashcard 19: Determine the derivative of $y = \text{arccos}(\text{cos}(x))$.

Answer: $-1$. Inverse function cancels on principal domain.

Flashcard 20: Evaluate the derivative of $\text{arccsc}(x)$ at $x = -2$.

Answer: $\frac{1}{2\text{sqrt}(3)}$. Substitute $x = -2$ into $-\frac{1}{|x|\sqrt{x^2-1}}$ formula.

Flashcard 21: Find the derivative of $y = \text{arccos}(\frac{x}{2})$ at $x = 1$.

Answer: $-\frac{1}{\sqrt{3}}$. Use chain rule with $\frac{d}{dx}[\frac{x}{2}] = \frac{1}{2}$ and evaluate at $x = 1$.

Flashcard 22: State the derivative of $\text{arcsec}(x)$.

Answer: $\frac{1}{|x| \sqrt{x^2 - 1}}$. Standard derivative formula for inverse secant function.

Flashcard 23: State the derivative of $\text{arccos}(x)$.

Answer: $-\frac{1}{\sqrt{1-x^2}}$. Standard derivative formula for inverse cosine function.

Flashcard 24: Identify the derivative of $\text{arccot}(8x)$ with respect to $x$.

Answer: $-\frac{8}{1+64x^2}$. Apply chain rule: derivative of $8x$ times $-\frac{1}{1+(8x)^2}$.

Flashcard 25: Identify the derivative of $\text{arcsec}(6x)$ with respect to $x$.

Answer: $\frac{6}{|6x| \sqrt{36x^2-1}}$ Apply chain rule with arcsec derivative formula.

Flashcard 26: Find the derivative of $y = \text{arccsc}(x^2)$ at $x = 2$.

Answer: $-\frac{1}{4\sqrt{3}}$. Use chain rule with $\frac{d}{dx}[x^2] = 2x$ and evaluate at $x = 2$.

Flashcard 27: What is the derivative of $\text{arcsec}(x)$ at $x = 2$?

Answer: $\frac{1}{2\text{sqrt}(3)}$. Substitute $x = 2$ into $\frac{1}{|x|\sqrt{x^2-1}}$ formula.

Flashcard 28: Determine the derivative of $y = \text{arcsin}(\text{sin}(x))$.

Answer: $1$. Inverse function cancels on principal domain.

Flashcard 29: Identify the derivative of $\text{arcsin}(2x)$ with respect to $x$.

Answer: $\frac{2}{\sqrt{1-4x^2}}$. Apply chain rule: derivative of $2x$ times $\frac{1}{\sqrt{1-(2x)^2}}$.

Flashcard 30: Identify the derivative of $\text{arccsc}(7x)$ with respect to $x$.

Answer: $-\frac{7}{|7x|\text{sqrt}(49x^2-1)}$. Apply chain rule with arccsc derivative formula.

Flashcard 31: Identify the derivative of $\text{arccos}(3x)$ with respect to $x$.

Answer: $-\frac{3}{\sqrt{1-9x^2}}$. Apply chain rule: derivative of $3x$ times $-\frac{1}{\sqrt{1-(3x)^2}}$.

Flashcard 32: Find the derivative of $y = \text{arccsc}(x^2)$ at $x = 2$.

Answer: $-\frac{1}{4\text{sqrt}(3)}$. Use chain rule with $\frac{d}{dx}[x^2] = 2x$ and evaluate at $x = 2$.

Flashcard 33: What is the derivative of $\text{arcsin}(x)$ at $x = 0$?

Answer: $1$. Substitute $x = 0$ into $\frac{1}{\sqrt{1-x^2}}$ formula.

Flashcard 34: Find the derivative of $y = \text{arctan}(x^3)$ at $x = 1$.

Answer: $3$. Use chain rule with $\frac{d}{dx}[x^3] = 3x^2$ and evaluate at $x = 1$.

Flashcard 35: Determine the derivative of $y = \text{arctan}(\text{tan}(x))$.

Answer: $1$. Inverse function cancels on principal domain.

Flashcard 36: What is the derivative of $\text{arctan}(x)$ at $x = 1$?

Answer: $\frac{1}{2}$. Substitute $x = 1$ into $\frac{1}{1+x^2}$ formula.

Flashcard 37: State the derivative of $\text{arctan}(x)$.

Answer: $\frac{1}{1+x^2}$. Standard derivative formula for inverse tangent function.

Flashcard 38: Evaluate the derivative of $y = \text{arcsin}(\frac{x}{2})$ at $x = 1$.

Answer: $\frac{1}{\sqrt{3}}$. Use chain rule with $\frac{d}{dx}[\frac{x}{2}] = \frac{1}{2}$ and evaluate at $x = 1$.

Flashcard 39: State the derivative of $\text{arcsec}(x)$.

Answer: $\frac{1}{|x| \sqrt{x^2 - 1}}$. Standard derivative formula for inverse secant function.

Flashcard 40: State the derivative of $\text{arccsc}(x)$.

Answer: $-\frac{1}{|x|\text{sqrt}(x^2-1)}$. Standard derivative formula for inverse cosecant function.

Flashcard 41: Find the derivative of $y = \text{arccot}(\frac{1}{x})$ at $x = 1$.

Answer: $-1$. Use chain rule with $\frac{d}{dx} \left( \frac{1}{x} \right) = -\frac{1}{x^2}$ and evaluate at $x = 1$.

Flashcard 42: State the derivative of $\text{arccos}(x)$.

Answer: -rac{1}{\sqrt{1 - x^2}}. Standard derivative formula for inverse cosine function.

Flashcard 43: State the derivative of $\text{arccot}(x)$.

Answer: $-\frac{1}{1+x^2}$. Standard derivative formula for inverse cotangent function.

Flashcard 44: What is the derivative of $\text{arcsin}(x)$ at $x = 0$?

Answer: $1$. Substitute $x = 0$ into $\frac{1}{\sqrt{1-x^2}}$ formula.

Flashcard 45: Determine the derivative of $y = \text{arctan}(\text{tan}(x))$.

Answer: $1$. Inverse function cancels on principal domain.

Flashcard 46: Identify the derivative of $\text{arcsin}(2x)$ with respect to $x$.

Answer: \frac{2}{\sqrt{1-4x^2)}. Apply chain rule: derivative of $2x$ times $\frac{1}{\sqrt{1-(2x)^2}}$.

Flashcard 47: Evaluate the derivative of $y = \text{arctan}(2x)$ at $x = 0.5$.

Answer: $\frac{2}{5}$. Use chain rule with $\frac{d}{dx}[2x] = 2$ and evaluate at $x = 0.5$.

Flashcard 48: Identify the derivative of $\text{arccot}(8x)$ with respect to $x$.

Answer: $-\frac{8}{1+64x^2}$. Apply chain rule: derivative of $8x$ times $-\frac{1}{1+(8x)^2}$.

Flashcard 49: State the derivative of $\text{arccot}(x)$.

Answer: $-\frac{1}{1+x^2}$. Standard derivative formula for inverse cotangent function.

Flashcard 50: Determine the derivative of $y = \text{arccot}(\text{cot}(x))$.

Answer: $-1$. Inverse function cancels on principal domain.

Flashcard 51: Determine the derivative of $y = \text{arccos}(\text{cos}(x))$.

Answer: $-1$. Inverse function cancels on principal domain.

Flashcard 52: Find the derivative of $y = \text{arccos}(\frac{x}{2})$ at $x = 1$.

Answer: $-\frac{1}{\sqrt{3}}$. Use chain rule with $\frac{d}{dx}[\frac{x}{2}] = \frac{1}{2}$ and evaluate at $x = 1$.

Flashcard 53: Find the derivative of $y = \text{arctan}(x^3)$ at $x = 1$.

Answer: $3$. Use chain rule with $\frac{d}{dx}[x^3] = 3x^2$ and evaluate at $x = 1$.

Flashcard 54: Identify the derivative of $\text{arccsc}(7x)$ with respect to $x$.

Answer: $-\frac{7}{|7x|\text{sqrt}(49x^2-1)}$. Apply chain rule with arccsc derivative formula.

Flashcard 55: Identify the derivative of $\text{arctan}(5x)$ with respect to $x$.

Answer: $\frac{5}{1+25x^2}$. Apply chain rule: derivative of $5x$ times $\frac{1}{1+(5x)^2}$.

Flashcard 56: What is the derivative of $\text{arcsec}(x)$ at $x = 2$?

Answer: $\frac{1}{2\sqrt{3}}$. Substitute $x = 2$ into $\frac{1}{|x|\sqrt{x^2-1}}$ formula.

Flashcard 57: Identify the derivative of $\text{arccos}(3x)$ with respect to $x$.

Answer: $-\frac{3}{\sqrt{1-9x^2}}$. Apply chain rule: derivative of $3x$ times $-\frac{1}{\sqrt{1-(3x)^2}}$.

Flashcard 58: Determine the derivative of $y = \text{arcsec}(\text{sec}(x))$.

Answer: $1$. Inverse function cancels on principal domain.

Flashcard 59: Determine the derivative of $y = \text{arccsc}(\text{csc}(x))$.

Answer: $-1$. Inverse function cancels on principal domain.

Flashcard 60: Evaluate the derivative of $y = \text{arcsin}(\frac{x}{2})$ at $x = 1$.

Answer: $\frac{1}{\sqrt{3}}$. Use chain rule with $\frac{d}{dx}[\frac{x}{2}] = \frac{1}{2}$ and evaluate at $x = 1$.