Differentiating Inverse Trigonometric Functions - AP Calculus AB

1. State the derivative of $\text{arccsc}(x)$.

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

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

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

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

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

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

   $1$. Inverse function cancels on principal domain.

5. What is the derivative of $\text{arctan}(x)$ at $x = 1$?

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

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

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

7. State the derivative of $\text{arcsin}(x)$.

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

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

   $1$. Inverse function cancels on principal domain.

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

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

10. State the derivative of $\text{arctan}(x)$.

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

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

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

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

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

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

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

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

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

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

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

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

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

17. What is the derivative of $\text{arcsin}(x)$ at $x = 0$?

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

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

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

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

    $1$. Inverse function cancels on principal domain.

20. State the derivative of $\text{arcsec}(x)$.

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

21. State the derivative of $\text{arccos}(x)$.

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

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

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

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

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

24. State the derivative of $\text{arccot}(x)$.

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

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

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

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

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

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

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

28. What is the derivative of $\text{arcsec}(x)$ at $x = 2$?

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

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

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

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

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

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