Differentiating Inverse Trigonometric Functions - AP Calculus BC

1. What is the chain rule application for $y = \sec^{-1}(g(x))$?

   $\frac{g'(x)}{|g(x)|\sqrt{(g(x))^2-1}}$. General chain rule pattern for inverse secant composition.

2. Find $\frac{d}{dx}(\sin^{-1}(2x))$.

   $\frac{2}{\sqrt{1-(2x)^2}}$. Chain rule with $\frac{d}{dx}(2x) = 2$ applied to $\sin^{-1}$ formula.

3. Find $\frac{d}{dx}(\cos^{-1}(4x^3))$.

   $-\frac{12x^2}{\sqrt{1-(4x^3)^2}}$. Chain rule with $\frac{d}{dx}(4x^3) = 12x^2$ applied to $\cos^{-1}$.

4. Find $\frac{d}{dx}(\tan^{-1}(\sqrt{3x}))$.

   $\frac{3}{2\sqrt{3x}(1+3x)}$. Chain rule with $\frac{d}{dx}(\sqrt{3x}) = \frac{3}{2\sqrt{3x}}$ applied to $\tan^{-1}$.

5. Find $\frac{d}{dx}(\cot^{-1}(\sqrt{2x}))$.

   $-\frac{1}{2\sqrt{2x}(1+2x)}$. Chain rule with $\frac{d}{dx}(\sqrt{2x}) = \frac{1}{\sqrt{2x}}$ applied to $\cot^{-1}$.

6. State the derivative of $\sin^{-1}(x)$.

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

7. Evaluate $\frac{d}{dx}(\cot^{-1}(x))$ at $x = 1$.

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

8. State the derivative of $\cos^{-1}(x)$.

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

9. Identify the derivative of $y = \cot^{-1}(\sqrt{x})$.

   $-\frac{1}{2\sqrt{x}(1+x)}$. Chain rule with $\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}$ applied to $\cot^{-1}$.

10. Find $\frac{d}{dx}(\csc^{-1}(2x^3))$.

    $-\frac{6x^2}{|2x^3|\sqrt{(2x^3)^2-1}}$. Chain rule with $\frac{d}{dx}(2x^3) = 6x^2$ applied to $\csc^{-1}$.

11. What is the chain rule application for $y = \cos^{-1}(g(x))$?

    $-\frac{g'(x)}{\sqrt{1-(g(x))^2}}$. General chain rule pattern for inverse cosine composition.

12. Identify the derivative of $y = \tan^{-1}(\sqrt{x})$.

    $\frac{1}{2\sqrt{x}(1+x)}$. Chain rule with $\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}$ applied to $\tan^{-1}$.

13. Evaluate $\frac{d}{dx}(\csc^{-1}(x))$ at $x = 2$.

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

14. Identify the derivative of $y = \sec^{-1}(x^4)$.

    $\frac{4x^3}{|x^4|\sqrt{x^8-1}}$. Chain rule with $\frac{d}{dx}(x^4) = 4x^3$ applied to $\sec^{-1}$ formula.

15. Identify the derivative of $y = \cos^{-1}(x^3)$.

    $-\frac{3x^2}{\sqrt{1-x^6}}$. Chain rule with $\frac{d}{dx}(x^3) = 3x^2$ applied to $\cos^{-1}$ formula.

16. Find $\frac{d}{dx}(x \cos^{-1}(x))$.

    $\cos^{-1}(x) - \frac{x}{\sqrt{1-x^2}}$. Product rule: $(uv)' = u'v + uv'$ applied to $x\cos^{-1}(x)$.

17. Find $\frac{d}{dx}(\cos^{-1}(3x))$.

    $-\frac{3}{\sqrt{1-(3x)^2}}$. Chain rule with $\frac{d}{dx}(3x) = 3$ applied to $\cos^{-1}$ formula.

18. Identify the derivative of $y = \sin^{-1}(x^2)$.

    $\frac{2x}{\sqrt{1-x^4}}$. Chain rule with $\frac{d}{dx}(x^2) = 2x$ applied to $\sin^{-1}$ formula.

19. Evaluate $\frac{d}{dx}(\tan^{-1}(x))$ at $x = 1$.

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

20. Evaluate $\frac{d}{dx}(\sin^{-1}(x))$ at $x = \frac{1}{2}$.

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

21. What is the chain rule application for $y = \csc^{-1}(g(x))$?

    $-\frac{g'(x)}{|g(x)|\sqrt{(g(x))^2-1}}$. General chain rule pattern for inverse cosecant composition.

22. Evaluate $\frac{d}{dx}(\cos^{-1}(x))$ at $x = \frac{1}{2}$.

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

23. Find $\frac{d}{dx}(\sec^{-1}(6x))$.

    $\frac{6}{|6x|\sqrt{(6x)^2-1}}$. Chain rule with $\frac{d}{dx}(6x) = 6$ applied to $\sec^{-1}$ formula.

24. Find $\frac{d}{dx}(x \sin^{-1}(x))$.

    $\sin^{-1}(x) + \frac{x}{\sqrt{1-x^2}}$. Product rule: $(uv)' = u'v + uv'$ applied to $x\sin^{-1}(x)$.

25. State the derivative of $\tan^{-1}(x)$.

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

26. Find $\frac{d}{dx}(\sec^{-1}(5x^2))$.

    $\frac{10x}{|5x^2|\sqrt{(5x^2)^2-1}}$. Chain rule with $\frac{d}{dx}(5x^2) = 10x$ applied to $\sec^{-1}$.

27. What is the chain rule application for $y = \tan^{-1}(g(x))$?

    $\frac{g'(x)}{1+(g(x))^2}$. General chain rule pattern for inverse tangent composition.

28. Find $\frac{d}{dx}(\sin^{-1}(3x^2))$.

    $\frac{6x}{\sqrt{1-(3x^2)^2}}$. Chain rule with $\frac{d}{dx}(3x^2) = 6x$ applied to $\sin^{-1}$.

29. Find $\frac{d}{dx}(\tan^{-1}(5x))$.

    $\frac{5}{1+(5x)^2}$. Chain rule with $\frac{d}{dx}(5x) = 5$ applied to $\tan^{-1}$ formula.

30. What is the chain rule application for $y = \cot^{-1}(g(x))$?

    $-\frac{g'(x)}{1+(g(x))^2}$. General chain rule pattern for inverse cotangent composition.

31. Find $\frac{d}{dx}(\cot^{-1}(4x))$.

    $-\frac{4}{1+(4x)^2}$. Chain rule with $\frac{d}{dx}(4x) = 4$ applied to $\cot^{-1}$ formula.

32. Identify the derivative of $y = \csc^{-1}(x^5)$.

    $-\frac{5x^4}{|x^5|\sqrt{x^{10}-1}}$. Chain rule with $\frac{d}{dx}(x^5) = 5x^4$ applied to $\csc^{-1}$ formula.

33. State the derivative of $\cot^{-1}(x)$.

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

34. What is the chain rule application for $y = \sin^{-1}(g(x))$?

    $\frac{g'(x)}{\sqrt{1-(g(x))^2}}$. General chain rule pattern for inverse sine composition.

35. State the derivative of $\sec^{-1}(x)$.

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

36. Find $\frac{d}{dx}(\csc^{-1}(7x))$.

    $-\frac{7}{|7x|\sqrt{(7x)^2-1}}$. Chain rule with $\frac{d}{dx}(7x) = 7$ applied to $\csc^{-1}$ formula.

37. Evaluate $\frac{d}{dx}(\sec^{-1}(x))$ at $x = 2$.

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

38. State the derivative of $\csc^{-1}(x)$.

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

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