Derivatives of Reciprocal Trig Functions - AP Calculus BC

1. Identify the derivative: $\frac{d}{dx} \sec(x)$

   $\sec(x)\tan(x)$. Standard derivative of secant.

2. Differentiate: $\cot(x)$

   $-\csc^2(x)$. Use derivative formula for $\cot(x)$.

3. Evaluate $\frac{d}{dx} \tan(x)$ at $x = 0$.

   $1$. $\sec^2(0) = 1^2 = 1$.

4. Find the derivative of $y = \cot(x)$.

   $\frac{dy}{dx} = -\csc^2(x)$. Apply cotangent derivative rule.

5. Find the derivative of $y = \tan(x)$.

   $\frac{dy}{dx} = \sec^2(x)$. Apply tangent derivative rule.

6. State the formula for the derivative of tangent.

   $\sec^2(x)$. Memorized derivative formula.

7. State the formula for the derivative of secant.

   $\sec(x)\tan(x)$. Memorized derivative formula.

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

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

9. What is the slope of the tangent line to $y = \tan(x)$ at $x = \frac{\pi}{4}$?

   $2$. Derivative equals slope at given point.

10. Determine the derivative of $f(x) = \cot^2(x)$.

    $-2\cot(x)\csc^2(x)$. Use chain rule: $2\cot(x) \cdot (-\csc^2(x))$.

11. Evaluate $\frac{d}{dx} \cot(x)$ at $x = \frac{\pi}{4}$.

    $-2$. $-\csc^2(\frac{\pi}{4}) = -(\sqrt{2})^2 = -2$.

12. What is the slope of the tangent line to $y = \cot(x)$ at $x = \frac{\pi}{2}$?

    $-1$. Derivative equals slope at given point.

13. Identify the derivative: $\frac{d}{dx} \csc(x)$

    $-\csc(x)\cot(x)$. Standard derivative of cosecant.

14. Find the derivative of $y = \csc(x)$.

    $\frac{dy}{dx} = -\csc(x)\cot(x)$. Apply cosecant derivative rule.

15. State the formula for the derivative of cosecant.

    $-\csc(x)\cot(x)$. Memorized derivative formula.

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

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

17. Compute the derivative of $f(x) = \sec(x)$ at $x = \frac{\pi}{3}$.

    $2\sqrt{3}$. $\sec(\frac{\pi}{3})\tan(\frac{\pi}{3}) = 2 \cdot \sqrt{3} = 2\sqrt{3}$.

18. Identify the derivative: $\frac{d}{dx} \tan(x)$

    $\sec^2(x)$. Standard derivative of tangent.

19. Evaluate $\frac{d}{dx} \sec(x)$ at $x = 0$.

    $0$. $\sec(0)\tan(0) = 1 \cdot 0 = 0$.

20. Differentiate: $\tan(x)$

    $\sec^2(x)$. Use derivative formula for $\tan(x)$.

21. Evaluate $\frac{d}{dx} \csc(x)$ at $x = \frac{\pi}{2}$.

    $0$. $-\csc(\frac{\pi}{2})\cot(\frac{\pi}{2}) = -1 \cdot 0 = 0$.

22. Identify the derivative: $\frac{d}{dx} \cot(x)$

    $-\csc^2(x)$. Standard derivative of cotangent.

23. Compute the derivative of $f(x) = \tan(x)$ at $x = \frac{\pi}{4}$.

    $2$. $\sec^2(\frac{\pi}{4}) = (\sqrt{2})^2 = 2$.

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

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

25. Determine the derivative of $f(x) = \tan^2(x)$.

    $2\tan(x)\sec^2(x)$. Use chain rule: $2\tan(x) \cdot \sec^2(x)$.

26. What is the derivative of $\cot(x)$?

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

27. Differentiate: $\csc(x)$

    $-\csc(x)\cot(x)$. Use derivative formula for $\csc(x)$.

28. State the formula for the derivative of cotangent.

    $-\csc^2(x)$. Memorized derivative formula.

29. Differentiate: $\sec(x)$

    $\sec(x)\tan(x)$. Use derivative formula for $\sec(x)$.

30. Find the derivative of $y = \sec(x)$.

    $\frac{dy}{dx} = \sec(x)\tan(x)$. Apply secant derivative rule.

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