Derivatives of Reciprocal Trig Functions - AP Calculus AB

1. Find the derivative of $h(x) = \csc(\ln(x))$.

   $-\frac{1}{x}\csc(\ln(x))\cot(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.

2. Determine $d/dx$ for $h(x) = \csc(2x)$.

   $-2\csc(2x)\cot(2x)$. Chain rule: multiply by derivative of inner function $2$.

3. Find $d/dx$ for $f(x) = \sec(\sin(x))$.

   $\cos(x)\sec(\sin(x))\tan(\sin(x))$. Chain rule: multiply by derivative of $\sin(x)$ which is $\cos(x)$.

4. What is the derivative of $f(x) = \cot(2x + 3)$?

   $-2\csc^2(2x + 3)$. Chain rule: multiply by derivative of inner function $2$.

5. Find $d/dx$ for $g(x) = \cot(\ln(x))$.

   $-\frac{1}{x}\csc^2(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.

6. Calculate $d/dx$ for $g(x) = \sec(x^5)$.

   $5x^4\sec(x^5)\tan(x^5)$. Chain rule: multiply by derivative of inner function $5x^4$.

7. Compute the derivative of $g(x) = \sec(5x)$.

   $5\sec(5x)\tan(5x)$. Chain rule: multiply by derivative of inner function $5$.

8. What is the derivative of $\tan(x)$?

   $\sec^2(x)$. Standard derivative formula for tangent function.

9. State the derivative of $g(x) = \cot(2x^3)$.

   $-6x^2\csc^2(2x^3)$. Chain rule: multiply by derivative of inner function $6x^2$.

10. State the derivative of $g(x) = \sec(x^3)$.

    $3x^2\sec(x^3)\tan(x^3)$. Chain rule: multiply by derivative of inner function $3x^2$.

11. Calculate the derivative of $y = \csc(4x)$.

    $-4\csc(4x)\cot(4x)$. Chain rule: multiply by derivative of inner function $4$.

12. Determine the derivative of $y = \csc(4x^2)$.

    $-8x\csc(4x^2)\cot(4x^2)$. Chain rule: multiply by derivative of inner function $8x$.

13. What is the derivative of $f(x) = \tan(x^2 + 1)$?

    $2x\sec^2(x^2 + 1)$. Chain rule: multiply by derivative of inner function $2x$.

14. State the derivative of $\cot(x)$.

    $-\csc^2(x)$. Standard derivative formula for cotangent function.

15. What is the derivative of $y = \tan(\sqrt{x})$?

    $\frac{1}{2\sqrt{x}}\sec^2(\sqrt{x})$. Chain rule: multiply by derivative of $\sqrt{x}$ which is $\frac{1}{2\sqrt{x}}$.

16. Find the derivative of $f(x) = \tan(x^2)$.

    $2x\sec^2(x^2)$. Chain rule: multiply by derivative of inner function $2x$.

17. Determine the derivative of $y = \tan(3x^2)$.

    $6x\sec^2(3x^2)$. Chain rule: multiply by derivative of inner function $6x$.

18. State $d/dx$ for $g(x) = \cot(\sqrt{x})$.

    $-\frac{1}{2\sqrt{x}}\csc^2(\sqrt{x})$. Chain rule: multiply by derivative of $\sqrt{x}$ which is $\frac{1}{2\sqrt{x}}$.

19. Compute $d/dx$ for $g(x) = \sec(x^2 + 1)$.

    $2x\sec(x^2 + 1)\tan(x^2 + 1)$. Chain rule: multiply by derivative of inner function $2x$.

20. Identify the derivative of $f(x) = \tan(2x + 1)$.

    $2\sec^2(2x + 1)$. Chain rule: multiply by derivative of inner function $2$.

21. What is the derivative of $f(x) = \cot(3x)$?

    $-3\csc^2(3x)$. Chain rule: multiply by derivative of inner function $3$.

22. State the derivative of $f(x) = \cot(3x^3)$.

    $-9x^2\csc^2(3x^3)$. Chain rule: multiply by derivative of inner function $9x^2$.

23. Find the derivative of $f(x) = \tan(e^x)$.

    $e^x\sec^2(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.

24. Identify the derivative of $y = \tan(x + \pi)$.

    $\sec^2(x + \pi)$. Chain rule: multiply by derivative of inner function $1$.

25. Determine the derivative of $y = \csc(x^2)$.

    $-2x\csc(x^2)\cot(x^2)$. Chain rule: multiply by derivative of inner function $2x$.

26. What is the derivative of $f(x) = \tan(\frac{1}{x})$?

    $-\frac{1}{x^2}\sec^2(\frac{1}{x})$. Chain rule: multiply by derivative of $\frac{1}{x}$ which is $-\frac{1}{x^2}$

27. What is the derivative of $y = \cot(x^3)$?

    $-3x^2\csc^2(x^3)$. Chain rule: multiply by derivative of inner function $3x^2$.

28. What is the derivative of $y = \cot(e^x)$?

    $-e^x\csc^2(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.

29. Find $d/dx$ for $h(x) = \sec(3x + 1)$.

    $3\sec(3x + 1)\tan(3x + 1)$. Chain rule: multiply by derivative of inner function $3$.

30. Compute the derivative of $g(x) = \csc(x^4)$.

    $-4x^3\csc(x^4)\cot(x^4)$. Chain rule: multiply by derivative of inner function $4x^3$.

31. What is the derivative of $f(x) = \tan(5x^2)$?

    $10x\sec^2(5x^2)$. Chain rule: multiply by derivative of inner function $10x$.

32. Compute the derivative of $h(x) = \sec(\ln(x))$.

    $\frac{1}{x}\sec(\ln(x))\tan(\ln(x))$. Chain rule: multiply by derivative of $\ln(x)$ which is $\frac{1}{x}$.

33. What is the derivative of $\sec(x)$?

    $\sec(x)\tan(x)$. Standard derivative formula for secant function.

34. Determine $d/dx$ for $y = \csc(e^x)$.

    $-e^x\csc(e^x)\cot(e^x)$. Chain rule: multiply by derivative of $e^x$ which is $e^x$.

35. Calculate the derivative of $h(x) = \sec(\sqrt{x})$.

    $\frac{1}{2\sqrt{x}}\sec(\sqrt{x})\tan(\sqrt{x})$. Chain rule: multiply by derivative of $\sqrt{x}$ which is $\frac{1}{2\sqrt{x}}$.

36. State the derivative of $\csc(x)$.

    $-\csc(x)\cot(x)$. Standard derivative formula for cosecant function.

37. Find $d/dx$ for $f(x) = \sec^2(x)$.

    $2\sec(x)\sec(x)\tan(x)$. Chain rule applied to $\sec^2(x) = 2\sec(x) \cdot \sec(x)\tan(x)$.

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