Derivatives of Trigonometry and Logarithmic Functions - AP Calculus AB

1. What is the derivative of $f(x) = \text{sin}(2x)$?

   $f'(x) = 2\text{cos}(2x)$. Apply chain rule: outer derivative times inner derivative.

2. What is the derivative of $f(x) = \text{cos}(x) + \text{ln}(x)$?

   $f'(x) = -\text{sin}(x) + \frac{1}{x}$. Apply linearity to find the sum of derivatives.

3. What is the second derivative of $\text{ln}(x)$?

   $-\frac{1}{x^2}$. Take the derivative of $\frac{1}{x}$ to get the second derivative.

4. What is the derivative of $\text{sin}(x)$?

   $\text{cos}(x)$. The derivative of sine is cosine.

5. What is the second derivative of $\text{sin}(x)$?

   $-\text{sin}(x)$. Take the derivative of $\text{cos}(x)$ to get the second derivative.

6. What is the derivative of $f(x) = \text{cos}(x) \times \text{ln}(x)$?

   $f'(x) = -\text{sin}(x)\text{ln}(x) + \frac{\text{cos}(x)}{x}$. Apply the product rule to both functions.

7. Evaluate the derivative of $f(x) = \text{ln}(x)$ at $x = 1$.

   $f'(1) = 1$. The derivative of $\text{ln}(x)$ is $\frac{1}{x}$, and $\frac{1}{1} = 1$.

8. Find the derivative of $f(x) = e^x + \text{ln}(x)$.

   $f'(x) = e^x + \frac{1}{x}$. Apply linearity to find the sum of individual derivatives.

9. Find the derivative of $f(x) = \text{ln}(x^2)$.

   $f'(x) = \frac{2}{x}$. Use logarithm property: $\text{ln}(x^2) = 2\text{ln}(x)$.

10. What is the derivative of $f(x) = \text{cos}(x) - e^x$?

    $f'(x) = -\text{sin}(x) - e^x$. Apply linearity: derivative of difference equals difference of derivatives.

11. What is the derivative of $\text{cos}(x)$?

    $-\text{sin}(x)$. The derivative of cosine is negative sine.

12. Find the derivative of $f(x) = e^{x^2}$ using chain rule.

    $f'(x) = 2xe^{x^2}$. Apply chain rule with outer function $e^u$ and inner $x^2$.

13. Evaluate the derivative at $x = 0$: $f(x) = \text{ln}(x + 1)$.

    $f'(0) = 1$. Use chain rule: derivative is $\frac{1}{x+1}$, and $\frac{1}{0+1} = 1$.

14. Evaluate the derivative at $x = \frac{\text{π}}{2}$: $f(x) = \text{sin}(x)$.

    $f'(\frac{\text{π}}{2}) = 0$. The derivative is $\text{cos}(x)$, and $\text{cos}(\frac{\pi}{2}) = 0$.

15. What is the second derivative of $\text{cos}(x)$?

    $-\text{cos}(x)$. Take the derivative of $-\text{sin}(x)$ to get the second derivative.

16. Find the derivative of $f(x) = -\text{sin}(x)$.

    $f'(x) = -\text{cos}(x)$. Factor out the negative sign from the derivative.

17. What is the derivative of $f(x) = \text{sin}(x) + e^x$?

    $f'(x) = \text{cos}(x) + e^x$. Apply linearity to find the sum of derivatives.

18. What is the derivative of $f(x) = \text{cos}^2(x)$?

    $f'(x) = -2\text{sin}(x)\text{cos}(x)$. Apply chain rule to the squared function.

19. Evaluate the derivative at $x = 0$: $f(x) = \text{cos}(x)$.

    $f'(0) = 0$. The derivative is $-\text{sin}(x)$, and $-\text{sin}(0) = 0$.

20. Evaluate the derivative at $x = 0$: $f(x) = e^x$.

    $f'(0) = 1$. The derivative of $e^x$ is $e^x$, and $e^0 = 1$.

21. Find the derivative of $f(x) = \text{ln}(2x)$.

    $f'(x) = \frac{1}{x}$. The derivative simplifies using logarithm properties.

22. Evaluate the derivative at $x = 0$: $f(x) = \ln(x + 1)$.

    $f'(0) = 1$. Use chain rule: derivative is $\frac{1}{x+1}$, and $\frac{1}{0+1} = 1$.

23. What is the derivative of $f(x) = \text{sin}(x) \times e^x$?

    $f'(x) = \text{cos}(x)e^x + \text{sin}(x)e^x$. Apply the product rule: $(uv)' = u'v + uv'$.

24. Find the derivative of $f(x) = e^x \times \text{ln}(x)$.

    $f'(x) = e^x \times \frac{1}{x} + \text{ln}(x) \times e^x$. Use the product rule on $e^x$ and $\text{ln}(x)$.

25. What is the derivative of $\text{ln}(x)$?

    $\frac{1}{x}$. The derivative of natural logarithm is the reciprocal.

26. What is the derivative of $f(x) = \text{cos}(2x)$?

    $f'(x) = -2\text{sin}(2x)$. Apply chain rule with inner function $2x$.

27. What is the derivative of $f(x) = \sin^2(x)$?

    $f'(x) = 2\sin(x)\cos(x)$. Apply chain rule to the squared function.

28. Find the third derivative of $f(x) = \text{cos}(x)$.

    $\text{sin}(x)$. The third derivative follows the cosine pattern: cos → -sin → -cos → sin.

29. Find the derivative of $f(x) = \text{sin}(x) + \text{cos}(x)$.

    $f'(x) = \text{cos}(x) - \text{sin}(x)$. Use linearity: derivative of sum equals sum of derivatives.

30. Find the derivative of $f(x) = e^{2x}$.

    $f'(x) = 2e^{2x}$. Use chain rule: derivative of $e^u$ is $e^u \cdot u'$.

31. Determine the derivative of $f(x) = 4e^x$.

    $f'(x) = 4e^x$. Use the constant multiple rule: multiply the derivative by 4.

32. Find the third derivative of $f(x) = \text{sin}(x)$.

    $-\text{cos}(x)$. The third derivative follows the sine pattern: sin → cos → -sin → -cos.

33. What is the derivative of $f(x) = \text{sin}^2(x)$?

    $f'(x) = 2\text{sin}(x)\text{cos}(x)$. Apply chain rule to the squared function.

34. Determine the derivative of $f(x) = 3\text{sin}(x)$.

    $f'(x) = 3\text{cos}(x)$. Use the constant multiple rule: multiply the derivative by 3.

35. What is the second derivative of $e^x$?

    $e^x$. The exponential function equals its own second derivative.

36. Find the derivative of $f(x) = -\text{cos}(x)$.

    $f'(x) = \text{sin}(x)$. Factor out the negative sign from the derivative.

37. What is the derivative of $f(x) = 2\text{ln}(x)$?

    $f'(x) = \frac{2}{x}$. Use the constant multiple rule with coefficient 2.

38. What is the derivative of $e^x$?

    $e^x$. The exponential function is its own derivative.

39. Evaluate the derivative at $x = 0$: $f(x) = \cos(x)$.

    $f'(0) = 0$. The derivative is $-\sin(x)$, and $-\sin(0) = 0$.

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