Applying the Power Rule - AP Calculus AB

1. Differentiate $f(x) = x^6 - x^2$ using the Power Rule.

   $f'(x) = 6x^5 - 2x$. Differentiate each term: $6x^5 - 2x$.

2. What is the derivative of $f(x) = -5x^{0.5}$?

   $f'(x) = -2.5x^{-0.5}$. Apply Power Rule: $-5 \cdot 0.5 \cdot x^{0.5-1} = -2.5x^{-0.5}$.

3. Find the derivative of $f(x) = x^0$.

   $f'(x) = 0$. $x^0 = 1$, which is constant, so derivative is 0.

4. What is the derivative of $f(x) = 8x^{-4}$?

   $f'(x) = -32x^{-5}$. Apply Power Rule: $8 \cdot (-4) \cdot x^{-4-1} = -32x^{-5}$.

5. State the Power Rule for differentiation.

   $\frac{d}{dx}[x^n] = nx^{n-1}$. Fundamental differentiation rule: bring down exponent, subtract 1 from power.

6. Differentiate $f(x) = 6x^3 + 7x$ using the Power Rule.

   $f'(x) = 18x^2 + 7$. Differentiate each term: $6 \cdot 3x^2 + 7 \cdot 1 = 18x^2 + 7$.

7. Find the derivative of $f(x) = 7x^{3/7}$ using the Power Rule.

   $f'(x) = 3x^{-4/7}$. Apply Power Rule: $7 \cdot \frac{3}{7} \cdot x^{\frac{3}{7}-1} = 3x^{-4/7}$.

8. Differentiate $f(x) = 11x^{-2/5}$ using the Power Rule.

   $f'(x) = -\frac{22}{5}x^{-7/5}$. Apply Power Rule: $11 \cdot (-\frac{2}{5}) \cdot x^{-\frac{2}{5}-1} = -\frac{22}{5}x^{-7/5}$.

9. Differentiate $f(x) = x^{1/3}$ using the Power Rule.

   $f'(x) = \frac{1}{3}x^{-2/3}$. Apply Power Rule: $\frac{1}{3} \cdot x^{\frac{1}{3}-1} = \frac{1}{3}x^{-2/3}$.

10. What is the derivative of $f(x) = x^{2/3}$?

    $f'(x) = \frac{2}{3}x^{-1/3}$. Apply Power Rule: $\frac{2}{3} \cdot x^{\frac{2}{3}-1} = \frac{2}{3}x^{-1/3}$.

11. Differentiate $f(x) = x^5$ using the Power Rule.

    $f'(x) = 5x^4$. Apply Power Rule: $5 \cdot x^{5-1} = 5x^4$.

12. Find the derivative of $f(x) = x^{3.5}$ using the Power Rule.

    $f'(x) = 3.5x^{2.5}$. Apply Power Rule: $3.5 \cdot x^{3.5-1} = 3.5x^{2.5}$.

13. Differentiate $f(x) = 3x^{5/2} - 4x^{3/2}$ using the Power Rule.

    $f'(x) = \frac{15}{2}x^{3/2} - 6x^{1/2}$. Differentiate each term using Power Rule.

14. Differentiate $f(x) = 15x^{1/4}$ using the Power Rule.

    $f'(x) = \frac{15}{4}x^{-3/4}$. Apply Power Rule: $15 \cdot \frac{1}{4} \cdot x^{\frac{1}{4}-1} = \frac{15}{4}x^{-3/4}$.

15. Differentiate $f(x) = 2x^4 + 3x^2$ using the Power Rule.

    $f'(x) = 8x^3 + 6x$. Differentiate each term: $2 \cdot 4x^3 + 3 \cdot 2x = 8x^3 + 6x$.

16. Differentiate $f(x) = x^{10}$ using the Power Rule.

    $f'(x) = 10x^9$. Apply Power Rule: $10 \cdot x^{10-1} = 10x^9$.

17. Find the derivative of $f(x) = x^{-5}$ using the Power Rule.

    $f'(x) = -5x^{-6}$. Apply Power Rule: $-5 \cdot x^{-5-1} = -5x^{-6}$.

18. Find $f'(x)$ for $f(x) = 10x^{-0.5}$. Use the Power Rule.

    $f'(x) = -5x^{-1.5}$. Apply Power Rule: $10 \cdot (-0.5) \cdot x^{-0.5-1} = -5x^{-1.5}$.

19. Differentiate $f(x) = 3x^{4.5}$ using the Power Rule.

    $f'(x) = 13.5x^{3.5}$. Apply Power Rule: $3 \cdot 4.5 \cdot x^{4.5-1} = 13.5x^{3.5}$.

20. What is the derivative of $f(x) = 7x^{-3}$?

    $f'(x) = -21x^{-4}$. Apply Power Rule: $7 \cdot (-3) \cdot x^{-3-1} = -21x^{-4}$.

21. Differentiate $f(x) = x^{1/2}$ using the Power Rule.

    $f'(x) = \frac{1}{2}x^{-1/2}$. Apply Power Rule: $\frac{1}{2} \cdot x^{\frac{1}{2}-1} = \frac{1}{2}x^{-1/2}$.

22. Differentiate $f(x) = x^{-2/3}$ using the Power Rule.

    $f'(x) = -\frac{2}{3}x^{-5/3}$. Apply Power Rule: $-\frac{2}{3} \cdot x^{-\frac{2}{3}-1} = -\frac{2}{3}x^{-5/3}$.

23. What is the derivative of $f(x) = x^9$?

    $f'(x) = 9x^8$. Apply Power Rule: $9 \cdot x^{9-1} = 9x^8$.

24. Differentiate $f(x) = x^{-3}$ using the Power Rule.

    $f'(x) = -3x^{-4}$. Apply Power Rule: $-3 \cdot x^{-3-1} = -3x^{-4}$.

25. Derive $f(x) = x^{-1}$ using the Power Rule.

    $f'(x) = -x^{-2}$. Apply Power Rule: $-1 \cdot x^{-1-1} = -x^{-2}$.

26. Find the derivative of $f(x) = x^{3.3}$ using the Power Rule.

    $f'(x) = 3.3x^{2.3}$. Apply Power Rule: $3.3 \cdot x^{3.3-1} = 3.3x^{2.3}$.

27. What is the derivative of $f(x) = 4x^0$?

    $f'(x) = 0$. $4x^0 = 4$, which is constant, so derivative is 0.

28. What is the derivative of $f(x) = 7x^2$?

    $f'(x) = 14x$. Apply Power Rule: $7 \cdot 2 \cdot x^{2-1} = 14x$.

29. What is the derivative of $f(x) = x^8$?

    $f'(x) = 8x^7$. Apply Power Rule: $8 \cdot x^{8-1} = 8x^7$.

30. What is the derivative of $f(x) = -2x^3$?

    $f'(x) = -6x^2$. Apply Power Rule: $-2 \cdot 3 \cdot x^{3-1} = -6x^2$.

31. Differentiate $f(x) = 9x^{-1/2}$ using the Power Rule.

    $f'(x) = -4.5x^{-3/2}$. Apply Power Rule: $9 \cdot (-\frac{1}{2}) \cdot x^{-\frac{1}{2}-1} = -4.5x^{-3/2}$.

32. Differentiate $f(x) = 5$ using the Power Rule.

    $f'(x) = 0$. Constant functions have derivative 0.

33. Differentiate $f(x) = 8x^{2/5}$ using the Power Rule.

    $f'(x) = \frac{16}{5}x^{-3/5}$. Apply Power Rule: $8 \cdot \frac{2}{5} \cdot x^{\frac{2}{5}-1} = \frac{16}{5}x^{-3/5}$.

34. Find the derivative of $f(x) = 4x^{-2} - 3x^{-4}$. Use Power Rule.

    $f'(x) = -8x^{-3} + 12x^{-5}$. Differentiate each term: $4(-2)x^{-3} - 3(-4)x^{-5}$.

35. Differentiate $f(x) = 5x^{7/4}$ using the Power Rule.

    $f'(x) = \frac{35}{4}x^{3/4}$. Apply Power Rule: $5 \cdot \frac{7}{4} \cdot x^{\frac{7}{4}-1} = \frac{35}{4}x^{3/4}$.

36. Differentiate $f(x) = 2x^5 + 3x^3 - x^2$ using the Power Rule.

    $f'(x) = 10x^4 + 9x^2 - 2x$. Differentiate each term using Power Rule.

37. Differentiate $f(x) = x^{0.1}$ using the Power Rule.

    $f'(x) = 0.1x^{-0.9}$. Apply Power Rule: $0.1 \cdot x^{0.1-1} = 0.1x^{-0.9}$.

38. Differentiate $f(x) = 13x^{-0.3}$ using the Power Rule.

    $f'(x) = -3.9x^{-1.3}$. Apply Power Rule: $13 \cdot (-0.3) \cdot x^{-0.3-1} = -3.9x^{-1.3}$.

39. Find the derivative of $f(x) = 6x^{-3/2}$ using the Power Rule.

    $f'(x) = -9x^{-5/2}$. Apply Power Rule: $6 \cdot (-\frac{3}{2}) \cdot x^{-\frac{3}{2}-1} = -9x^{-5/2}$.

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