AP Calculus AB Flashcards: Differentiating Inverse Functions

What this deck covers

This deck focuses on Differentiating Inverse 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: Find the derivative of $y = \text{arcsin}(2x)$ at $x = 0$.

Answer: $2$. Using chain rule: $\frac{d}{dx}[\arcsin(2x)] = \frac{2}{\sqrt{1-(2x)^2}}$, at $x=0$ gives $2$.

Flashcard 2: Find $(f^{-1})'(2)$ if $f(x) = e^x$ and $f(a) = 2$.

Answer: $\frac{1}{2}$. Since $f(\ln 2) = 2$ and $f'(\ln 2) = 2$, so $(f^{-1})'(2) = \frac{1}{2}$.

Flashcard 3: Find the derivative of $y = \text{arccos}(5x)$ at $x = 0$.

Answer: $-5$. Using chain rule: $\frac{d}{dx}[\arccos(5x)] = -\frac{5}{\sqrt{1-(5x)^2}}$, at $x=0$ gives $-5$.

Flashcard 4: Which inverse function has a derivative of $-\frac{1}{1+x^2}$?

Answer: $\text{arccot}(x)$. Inverse cotangent has derivative $-\frac{1}{1+x^2}$.

Flashcard 5: Find the derivative of $y = \text{arcsin}(2x)$ at $x = 0$.

Answer: $2$. Using chain rule: $\frac{d}{dx}[\arcsin(2x)] = \frac{2}{\sqrt{1-(2x)^2}}$, at $x=0$ gives $2$.

Flashcard 6: Find $(f^{-1})'(b)$ if $f(x) = x^3 + x$ and $f(a) = b$.

Answer: $\frac{1}{3a^2+1}$. Using inverse function theorem: $(f^{-1})'(b) = \frac{1}{f'(a)} = \frac{1}{3a^2+1}$.

Flashcard 7: Determine if $y = \text{arcsin}(x)$ is differentiable at $x = 1$.

Answer: No, $y = \text{arcsin}(x)$ is not differentiable at $x = 1$. At domain boundary, derivative is undefined.

Flashcard 8: Which inverse function has a derivative of $\frac{1}{1+x^2}$?

Answer: $\text{arctan}(x)$. Inverse tangent has derivative $\frac{1}{1+x^2}$.

Flashcard 9: What is the domain of $y = \text{arccos}(x)$?

Answer: $[-1, 1]$. Domain restricted to values where $-1 \leq x \leq 1$.

Flashcard 10: What is the derivative of $y = \text{arctan}(x)$?

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

Flashcard 11: Determine if $y = \text{arcsec}(x)$ is differentiable at $x = 0.5$.

Answer: No, $y = \text{arcsec}(x)$ is not differentiable at $x = 0.5$. Domain of $\arcsec$ is $|x| \geq 1$, so $x = 0.5$ is outside domain.

Flashcard 12: Which inverse function has a derivative of $\frac{1}{1+x^2}$?

Answer: $\text{arctan}(x)$. Inverse tangent has derivative $\frac{1}{1+x^2}$.

Flashcard 13: What is the derivative of $y = \text{arccot}(x)$?

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

Flashcard 14: What is the derivative of $y = \text{arccot}(x)$?

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

Flashcard 15: Find the derivative of $y = \text{arctan}(3x)$ at $x = 0$.

Answer: $3$. Using chain rule: $\frac{d}{dx}[\arctan(3x)] = \frac{3}{1+(3x)^2}$, at $x=0$ gives $3$.

Flashcard 16: What is the domain of $y = \text{arccos}(x)$?

Answer: $[-1, 1]$. Domain restricted to values where $-1 \leq x \leq 1$.

Flashcard 17: What is the domain of $y = \text{arcsin}(x)$?

Answer: $[-1, 1]$. Domain restricted to values where $-1 \leq x \leq 1$.

Flashcard 18: Find the derivative of $y = \text{arccot}(6x)$ at $x = 0$.

Answer: $-6$. Using chain rule: $\frac{d}{dx}[\arccot(6x)] = -\frac{6}{1+(6x)^2}$, at $x=0$ gives $-6$.

Flashcard 19: Which inverse function has a derivative of $-\frac{1}{1+x^2}$?

Answer: $\text{arccot}(x)$. Inverse cotangent has derivative $-\frac{1}{1+x^2}$.

Flashcard 20: Find $(f^{-1})'(b)$ if $f(x) = x^3 + x$ and $f(a) = b$.

Answer: $\frac{1}{3a^2+1}$. Using inverse function theorem: $(f^{-1})'(b) = \frac{1}{f'(a)} = \frac{1}{3a^2+1}$.

Flashcard 21: Determine if $y = \text{arcsin}(x)$ is differentiable at $x = 1$.

Answer: No, $y = \text{arcsin}(x)$ is not differentiable at $x = 1$. At domain boundary, derivative is undefined.

Flashcard 22: Find the derivative of $y = \text{arccos}(5x)$ at $x = 0$.

Answer: $-5$. Using chain rule: $\frac{d}{dx}[\arccos(5x)] = -\frac{5}{\sqrt{1-(5x)^2}}$, at $x=0$ gives $-5$.

Flashcard 23: Find $(f^{-1})'(2)$ if $f(x) = e^x$ and $f(a) = 2$.

Answer: $\frac{1}{2}$. Since $f(\ln 2) = 2$ and $f'(\ln 2) = 2$, so $(f^{-1})'(2) = \frac{1}{2}$.

Flashcard 24: Find the derivative of $y = \arccot(6x)$ at $x = 0$.

Answer: $-6$. Using chain rule: $\frac{d}{dx}[\arccot(6x)] = -\frac{6}{1+(6x)^2}$, at $x=0$ gives $-6$.

Flashcard 25: State the formula for the derivative of an inverse function.

Answer: $(f^{-1})'(b) = \frac{1}{f'(a)}$ where $f(a) = b$. The inverse function derivative theorem.

Flashcard 26: What is the domain of $y = \text{arcsin}(x)$?

Answer: $[-1, 1]$. Domain restricted to values where $-1 \leq x \leq 1$.

Flashcard 27: Find the derivative of $y = \text{arctan}(3x)$ at $x = 0$.

Answer: $3$. Using chain rule: $\frac{d}{dx}[\arctan(3x)] = \frac{3}{1+(3x)^2}$, at $x=0$ gives $3$.

Flashcard 28: Determine if $y = \text{arcsec}(x)$ is differentiable at $x = 0.5$.

Answer: No, $y = \text{arcsec}(x)$ is not differentiable at $x = 0.5$. Domain of $\arcsec$ is $|x| \geq 1$, so $x = 0.5$ is outside domain.

Flashcard 29: State the formula for the derivative of an inverse function.

Answer: $(f^{-1})'(b) = \frac{1}{f'(a)}$ where $f(a) = b$. The inverse function derivative theorem.

Flashcard 30: What is the derivative of $y = \text{arctan}(x)$?

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