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: 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 2: 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 3: 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 4: 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 5: Determine the derivative of $y = \text{arcsec}(\text{sec}(x))$.

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

Flashcard 6: Determine the derivative of $y = \text{arcsin}(\text{sin}(x))$.

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

Flashcard 7: 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 8: State the derivative of $\text{arcsec}(x)$.

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

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

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

Flashcard 10: 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 11: 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 12: 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 13: Determine the derivative of $y = \text{arccot}(\text{cot}(x))$.

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

Flashcard 14: Determine the derivative of $y = \text{arctan}(\text{tan}(x))$.

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

Flashcard 15: 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 16: State the derivative of $\text{arccot}(x)$.

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

Flashcard 17: 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 18: 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 19: 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 20: 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 21: State the derivative of $\text{arcsin}(x)$.

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

Flashcard 22: 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 23: State the derivative of $\text{arccsc}(x)$.

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

Flashcard 24: 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 25: Determine the derivative of $y = \text{arccos}(\text{cos}(x))$.

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

Flashcard 26: 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 27: 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 28: Determine the derivative of $y = \text{arccsc}(\text{csc}(x))$.

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

Flashcard 29: 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 30: State the derivative of $\text{arccos}(x)$.

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