Derivatives of Trigonometry and Logarithmic Functions - AP Calculus BC

1. Find the derivative: $y = \text{ln } x$.

   $\frac{dy}{dx} = \frac{1}{x}$. Apply the derivative rule for $\text{ln } x$.

2. Find the second derivative of $y = \text{ln } x$.

   $\frac{d^2y}{dx^2} = -\frac{1}{x^2}$. Differentiate $f'(x) = \frac{1}{x}$ to get $f''(x) = -\frac{1}{x^2}$.

3. Find the derivative of $f(x) = \text{ln }(5x)$.

   $f'(x) = \frac{1}{x}$. Constant multiple rule: $\frac{d}{dx}[\text{ln}(5x)] = \frac{1}{5x} \cdot 5 = \frac{1}{x}$.

4. Find the second derivative of $y = e^x$.

   $\frac{d^2y}{dx^2} = e^x$. Differentiate $f'(x) = e^x$ to get $f''(x) = e^x$.

5. Determine the derivative of $f(x) = xe^x$.

   $f'(x) = e^x + xe^x$. Use product rule: $(uv)' = u'v + uv'$.

6. Determine $f'(x)$ if $f(x) = \text{cos } x + \text{sin } x$.

   $f'(x) = -\text{sin } x + \text{cos } x$. Use sum rule: derivative of each term separately.

7. Differentiate $f(x) = \text{sin } x$ at $x = \frac{\text{π}}{2}$.

   $f'\big(\frac{\text{π}}{2}\big) = 0$. $f'(x) = \text{cos } x$, so $f'\big(\frac{\text{π}}{2}\big) = \text{cos}\big(\frac{\text{π}}{2}\big) = 0$.

8. What is the derivative of $\text{sin } x$?

   $\text{cos } x$. Basic derivative rule for sine function.

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

   $f'(x) = 2e^{2x}$. Use chain rule with inner function $2x$.

10. Find the derivative of $f(x) = \text{cos }(3x)$.

    $f'(x) = -3\text{sin }(3x)$. Use chain rule with inner function $3x$.

11. Find the derivative: $y = \text{cos } x$.

    $\frac{dy}{dx} = -\text{sin } x$. Apply the derivative rule for $\text{cos } x$.

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

    $f'(x) = 2\text{cos }(2x)$. Use chain rule with inner function $2x$.

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

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

14. Determine the derivative of $f(x) = x\text{cos } x$.

    $f'(x) = \text{cos } x - x\text{sin } x$. Use product rule: $(uv)' = u'v + uv'$.

15. Find the derivative: $y = e^x$.

    $\frac{dy}{dx} = e^x$. Apply the derivative rule for $e^x$.

16. Determine $f'(x)$ if $f(x) = \cos x + \sin x$.

    $f'(x) = -\sin x + \cos x$. Use sum rule: derivative of each term separately.

17. Evaluate the derivative: $f(x) = \text{ln } x$ at $x = 2$.

    $f'(2) = \frac{1}{2}$. $f'(x) = \frac{1}{x}$, so $f'(2) = \frac{1}{2}$.

18. Determine $f'(x)$ if $f(x) = e^x + \text{ln } x$.

    $f'(x) = e^x + \frac{1}{x}$. Use sum rule: derivative of each term separately.

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

    $f'(0) = 1$. $f'(x) = e^x$, so $f'(0) = e^0 = 1$.

20. What is the derivative of $\text{ln } x$?

    $\frac{1}{x}$. Standard derivative of natural logarithm.

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

    $f'(x) = \text{ln } x + 1$. Use product rule: $(uv)' = u'v + uv'$.

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

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

23. Differentiate $f(x) = 3\text{cos } x - 2\text{sin } x$.

    $f'(x) = -3\text{sin } x - 2\text{cos } x$. Use sum/difference rule with constant multiples.

24. Evaluate $f'(x)$ for $f(x) = e^x - \text{ln } x$ at $x = 1$.

    $f'(1) = e - 1$. $f'(x) = e^x - \frac{1}{x}$, so $f'(1) = e - 1$.

25. Evaluate the derivative: $f(x) = e^x$ at $x = \text{ln } 2$.

    $f'(\text{ln } 2) = 2$. $f'(x) = e^x$, so $f'(\text{ln } 2) = e^{\text{ln } 2} = 2$.

26. Find the slope of the tangent to $y = \text{ln } x$ at $x = 4$.

    Slope = $\frac{1}{4}$. The slope equals the derivative at the point.

27. What is the derivative of $\text{cos } x$?

    $-\text{sin } x$. Basic derivative rule for cosine function.

28. Differentiate $f(x) = \text{cos } x$ at $x = 0$.

    $f'(0) = 0$. $f'(x) = -\text{sin } x$, so $f'(0) = -\text{sin }(0) = 0$.

29. Differentiate $f(x) = 5e^x + 7\text{ln } x$.

    $f'(x) = 5e^x + \frac{7}{x}$. Use sum rule with constant multiples.

30. Find the derivative of $f(x) = \sin(2x)$.

    $f'(x) = 2\cos(2x)$. Use chain rule with inner function $2x$.

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

    $f'(0) = 1$. $f'(x) = \text{cos } x$, so $f'(0) = \text{cos}(0) = 1$.

32. Evaluate $f'(x)$ for $f(x) = \text{cos } x - e^x$ at $x = 0$.

    $f'(0) = -1$. $f'(x) = -\text{sin } x - e^x$, so $f'(0) = 0 - 1 = -1$.

33. Find the second derivative of $y = \text{cos } x$.

    $\frac{d^2y}{dx^2} = -\text{cos } x$. Differentiate $f'(x) = -\text{sin } x$ to get $f''(x) = -\text{cos } x$.

34. Find the second derivative of $y = \text{sin } x$.

    $\frac{d^2y}{dx^2} = -\text{sin } x$. Differentiate $f'(x) = \text{cos } x$ to get $f''(x) = -\text{sin } x$.

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

    $f'(1) = 1$. $f'(x) = \frac{1}{x}$, so $f'(1) = \frac{1}{1} = 1$.

36. Find the derivative: $y = \text{sin } x$.

    $\frac{dy}{dx} = \text{cos } x$. Apply the derivative rule for $\text{sin } x$.

37. Differentiate $f(x) = e^x$ at $x = 1$.

    $f'(1) = e$. $f'(x) = e^x$, so $f'(1) = e^1 = e$.

38. Differentiate $f(x) = \text{ln } x$ at $x = e$.

    $f'(e) = \frac{1}{e}$. $f'(x) = \frac{1}{x}$, so $f'(e) = \frac{1}{e}$.

39. Find the slope of the tangent to $y = e^x$ at $x = 2$.

    Slope = $e^2$. The slope equals the derivative at the point.

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

    $f'\big(\frac{\text{π}}{2}\big) = -1$. $f'(x) = -\text{sin } x$, so $f'\big(\frac{\text{π}}{2}\big) = -\text{sin}\big(\frac{\text{π}}{2}\big) = -1$.

Study these flashcards