AP Calculus BC 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 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: Identify the derivative of $y = \text{arctan}(x)$.

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

Flashcard 2: Which theorem relates derivatives of inverses to the original function?

Answer: Inverse Function Theorem. States that $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$ when $f'(f^{-1}(x)) \neq 0$.

Flashcard 3: Identify the derivative of $y = \text{arccot}(x)$.

Answer: $-\frac{1}{1+x^2}$. Derivative of inverse cotangent, negative of arctan derivative.

Flashcard 4: What is the derivative of $y = \text{ln}|x|$?

Answer: $\frac{1}{x}$ for $x \neq 0$. Chain rule applied to absolute value inside logarithm.

Flashcard 5: State the derivative formula for the inverse function of $f(x)$.

Answer: $\frac{1}{f'(f^{-1}(x))}$. Apply the Inverse Function Theorem: $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$.

Flashcard 6: What is the derivative of $y = \text{ln}(x)$?

Answer: $\frac{1}{x}$. Standard derivative of natural logarithm function.

Flashcard 7: Find $(f^{-1})'(2)$ if $f(x)=x^3+1$ and $f^{-1}(2)=1$.

Answer: $(f^{-1})'(2)=\frac{1}{3}$. Since $f'(x)=3x^2$ and $f'(1)=3$, reciprocal is $\frac{1}{3}$.

Flashcard 8: State the derivative formula for an inverse function: what is $(f^{-1})'(x)$ in terms of $f'$?

Answer: $(f^{-1})'(x)=\frac{1}{f'(f^{-1}(x))}$. Derivative of inverse is reciprocal of original derivative at corresponding point.

Flashcard 9: Find $(f^{-1})'(1)$ if $f(x)=e^x$.

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

Flashcard 10: Find $(f^{-1})'(e)$ if $f(x)=\ln(x)$.

Answer: $(f^{-1})'(e)=e$. Since $e^x$ and $\ln(x)$ are inverses, $f'(1)=\frac{1}{e}$ gives reciprocal $e$.

Flashcard 11: Find $(f^{-1})'(\frac{\pi}{2})$ if $f(x)=\sin(x)$ with domain $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$.

Answer: Undefined because $\cos\left(\frac{\pi}{2}\right)=0$. At $x=\frac{\pi}{2}$, $\sin'(x)=\cos(x)=0$, so undefined.

Flashcard 12: Find $(f^{-1})'(2)$ if $f(x)=x^2+1$ with domain $[0,\infty)$.

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

Flashcard 13: Identify the correct expression: which equals $(f^{-1})'(x)$, $\frac{1}{f'(f^{-1}(x))}$ or $\frac{1}{f'(x)}$?

Answer: $\frac{1}{f'(f^{-1}(x))}$. Must evaluate derivative at $f^{-1}(x)$, not at $x$ directly.

Flashcard 14: Find $(f^{-1})'(8)$ if $f(x)=x^3$ and $f^{-1}(8)=2$.

Answer: $(f^{-1})'(8)=\frac{1}{12}$. Since $f'(x)=3x^2$ and $f'(2)=12$, reciprocal is $\frac{1}{12}$.

Flashcard 15: What is the inverse-derivative relationship between slopes: how are $f'(a)$ and $(f^{-1})'(b)$ related when $f(a)=b$?

Answer: $f'(a)\cdot (f^{-1})'(b)=1$. Slopes of inverse functions multiply to 1 at corresponding points.

Flashcard 16: What is the equivalent point-slope form for inverses: if $f(a)=b$, what is $(f^{-1})'(b)$?

Answer: $(f^{-1})'(b)=\frac{1}{f'(a)}$. When $f(a)=b$, slopes at $(a,b)$ and $(b,a)$ are reciprocals.

Flashcard 17: State the chain-rule identity used for inverse differentiation: what is $f(f^{-1}(x))$?

Answer: $f(f^{-1}(x))=x$. Composing a function with its inverse yields the identity function.

Flashcard 18: Identify the graph transformation for inverses: across which line are $y=f(x)$ and $y=f^{-1}(x)$ reflected?

Answer: Reflection across $y=x$. Inverse functions are mirror images across the diagonal line.

Flashcard 19: What is the geometric meaning of inverse derivatives: how do the tangent line slopes compare at inverse points?

Answer: Slopes are reciprocals at $(a,b)$ and $(b,a)$. Tangent lines at $(a,b)$ and $(b,a)$ have reciprocal slopes.

Flashcard 20: State the chain-rule identity used for inverse differentiation: what is $f^{-1}(f(x))$ (on the domain of invertibility)?

Answer: $f^{-1}(f(x))=x$. Inverse followed by function returns original input on valid domain.

Flashcard 21: Identify the key condition needed to differentiate an inverse: what must be true about $f'(a)$?

Answer: $f'(a)\ne 0$. Inverse derivative exists only when original derivative is nonzero.

Flashcard 22: Find $(f^{-1})'(4)$ given $f(2)=4$ and $f'(2)=5$.

Answer: $(f^{-1})'(4)=\frac{1}{5}$. Since $f(2)=4$ and $f'(2)=5$, use reciprocal formula.

Flashcard 23: Find $(f^{-1})'(0)$ given $f(-3)=0$ and $f'(-3)=-2$.

Answer: $(f^{-1})'(0)=-\frac{1}{2}$. Since $f(-3)=0$ and $f'(-3)=-2$, take reciprocal.

Flashcard 24: Find $(f^{-1})'(1)$ given $f(0)=1$ and $f'(0)=\frac{1}{4}$.

Answer: $(f^{-1})'(1)=4$. Since $f(0)=1$ and $f'(0)=\frac{1}{4}$, reciprocal is $4$.

Flashcard 25: Find $(f^{-1})'(9)$ given $f(3)=9$ and $f'(3)=0$.

Answer: Undefined because $f'(3)=0$. Cannot divide by zero when $f'(3)=0$.

Flashcard 26: Identify the derivative of $y = \text{arccot}(x)$.

Answer: $-\frac{1}{1+x^2}$. Derivative of inverse cotangent, negative of arctan derivative.

Flashcard 27: What is the derivative of $y = \text{ln}(x)$?

Answer: $\frac{1}{x}$. Standard derivative of natural logarithm function.

Flashcard 28: State the derivative formula for the inverse function of $f(x)$.

Answer: $\frac{1}{f'(f^{-1}(x))}$. Apply the Inverse Function Theorem: $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$.

Flashcard 29: What is the derivative of $y = \text{ln}|x|$?

Answer: $\frac{1}{x}$ for $x \neq 0$. Chain rule applied to absolute value inside logarithm.

Flashcard 30: Identify the derivative of $y = \text{arctan}(x)$.

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