Differentiating Inverse Functions - AP Calculus AB

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

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

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

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

3. Which inverse function has a derivative of $-\frac{1}{1+x^2}$?

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

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

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

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

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

6. Which inverse function has a derivative of $\frac{1}{1+x^2}$?

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

7. What is the domain of $y = \text{arccos}(x)$?

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

8. What is the derivative of $y = \text{arctan}(x)$?

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

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

   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.

10. What is the derivative of $y = \text{arccot}(x)$?

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

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

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

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

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

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

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

14. What is the domain of $y = \text{arcsin}(x)$?

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

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

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

16. State the formula for the derivative of an inverse function.

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

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