Calculating Higher-Order Derivatives - AP Calculus AB

1. What is the second derivative of $f(x) = x^3$?

   $f''(x) = 6x$. Apply power rule twice: $f'(x) = 3x^2$, then $f''(x) = 6x$.

2. Calculate the second derivative of $f(x) = \frac{1}{4}x^4 + x$.

   $f''(x) = 3x^2$. First derivative is $x^3 + 1$, second derivative is $3x^2$.

3. What is the second derivative of $f(x) = x^2 + 2x + 1$?

   $f''(x) = 2$. First derivative is $2x + 2$, second derivative is constant $2$.

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

   $f''(x) = 4e^{2x}$. Use chain rule: $f'(x) = 2e^{2x}$, then $f''(x) = 4e^{2x}$.

5. State the second derivative of $f(x) = \frac{1}{2}x^2 - x$.

   $f''(x) = 1$. First derivative is $x - 1$, second derivative is constant $1$.

6. Find the second derivative of $f(x) = x^3 - 3x^2 + 3x - 1$.

   $f''(x) = 6x - 6$. First derivative is $3x^2 - 6x + 3$, second derivative is $6x - 6$.

7. Calculate the third derivative of $f(x) = x^5 - x^3 + x$.

   $f^{(3)}(x) = 60x^2 - 6$. Differentiate three times: $f'(x) = 5x^4 - 3x^2 + 1$, then continue.

8. Calculate the second derivative of $f(x) = \frac{1}{x}$.

   $f''(x) = \frac{2}{x^3}$. Rewrite as $x^{-1}$, apply power rule: $f'(x) = -x^{-2}$, $f''(x) = 2x^{-3}$.

9. Calculate the second derivative of $f(x) = \frac{1}{3}x^3 + 2x$.

   $f''(x) = 2x$. First derivative is $x^2 + 2$, second derivative is $2x$.

10. State the second derivative of $f(x) = \frac{1}{2}x^2 + 3x + 5$.

    $f''(x) = 1$. First derivative is $x + 3$, second derivative is constant $1$.

11. Find the second derivative of $f(x) = \frac{1}{3}x^3$.

    $f''(x) = 2x$. First derivative is $x^2$, second derivative is $2x$.

12. State the second derivative of $f(x) = \frac{1}{3}x^3 - x^2 + 1$.

    $f''(x) = 2x - 2$. First derivative is $x^2 - 2x$, second derivative is $2x - 2$.

13. What is the second derivative of $f(x) = e^x$?

    $f''(x) = e^x$. The exponential function $e^x$ is its own derivative.

14. What is the third derivative of $f(x) = \frac{1}{3}x^3+x^2$?

    $f^{(3)}(x) = 2$. First derivative is $x^2 + 2x$, second is $2x + 2$, third is $2$.

15. Find the second derivative of $f(x) = x^5 + x^4 + x^3$.

    $f''(x) = 20x^3 + 12x^2 + 6x$. Apply power rule twice to each term separately.

16. What is the second derivative of $f(x) = \frac{1}{2}x^2$?

    $f''(x) = 1$. First derivative is $x$, second derivative is constant $1$.

17. What is the third derivative of $f(x) = 7x^5 - 3x^4 + x^3$?

    $f^{(3)}(x) = 420x^2 - 72x + 6$. Differentiate each term three times using power rule.

18. Calculate the third derivative of $f(x) = \frac{1}{6}x^6$.

    $f^{(3)}(x) = 20x^3$. Apply power rule three times: $f'(x) = x^5$, $f''(x) = 5x^4$, $f'''(x) = 20x^3$.

19. State the second derivative of $f(x) = \frac{1}{4}x^4 - x^2$.

    $f''(x) = 3x^2 - 2$. Apply power rule twice: $f'(x) = x^3 - 2x$, then $f''(x) = 3x^2 - 2$.

20. Find the third derivative of $f(x) = 5x^4 + 3x^2$.

    $f^{(3)}(x) = 120x$. Differentiate three times: $f'(x) = 20x^3 + 6x$, $f''(x) = 60x^2 + 6$, $f'''(x) = 120x$.

21. What is the second derivative of $f(x) = x^2 - 4x + 4$?

    $f''(x) = 2$. First derivative is $2x - 4$, second derivative is constant $2$.

22. What is the second derivative of $f(x) = \frac{1}{5}x^5$?

    $f''(x) = 4x^3$. Apply power rule twice: $f'(x) = x^4$, then $f''(x) = 4x^3$.

23. What is the second derivative of $f(x) = x^3 + 3x^2 + 3x + 1$?

    $f''(x) = 6x + 6$. First derivative is $3x^2 + 6x + 3$, second derivative is $6x + 6$.

24. Find the second derivative of $f(x) = e^{3x}$.

    $f''(x) = 9e^{3x}$. Use chain rule: $f'(x) = 3e^{3x}$, then $f''(x) = 9e^{3x}$.

25. Calculate the second derivative of $f(x) = 4x^4 - 2x^2$.

    $f''(x) = 48x^2 - 4$. Apply power rule twice: $f'(x) = 16x^3 - 4x$, then $f''(x) = 48x^2 - 4$.

26. Calculate the second derivative of $f(x) = 3x^3 - x^2 + x$.

    $f''(x) = 18x - 2$. First derivative is $9x^2 - 2x + 1$, second derivative is $18x - 2$.

27. Identify the second derivative of $f(x) = \frac{1}{2}x^2 - x + 1$.

    $f''(x) = 1$. First derivative is $x - 1$, second derivative is constant $1$.

28. What is the second derivative of $f(x) = x^4 - 2x^2 + 1$?

    $f''(x) = 12x^2 - 4$. First derivative is $4x^3 - 4x$, second derivative is $12x^2 - 4$.

29. State the formula for the $n$-th derivative of $f(x) = x^n$.

    $f^{(n)}(x) = n!$ if $n$ is a positive integer. Each differentiation reduces the power by 1 and multiplies by the current power.

30. Calculate the second derivative of $f(x) = 3x^3 - 3x^2 + 3x - 1$.

    $f''(x) = 18x - 6$. First derivative is $9x^2 - 6x + 3$, second derivative is $18x - 6$.

31. What is the second derivative of $f(x) = \frac{1}{5}x^5 - \frac{1}{3}x^3$?

    $f''(x) = 4x^3 - 2x$. Apply power rule twice to each term separately.

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