Applying the Power Rule - AP Calculus BC

1. Differentiate: $f(x) = 5x^{3.5}$.

   $f'(x) = 17.5x^{2.5}$. Factor out 5, apply power rule: $5 \cdot 3.5x^{2.5}$.

2. Find the derivative of $x^{\frac{1}{2}}$.

   $\frac{1}{2}x^{-\frac{1}{2}}$. Power rule: $\frac{1}{2} \cdot x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}}$.

3. Differentiate: $f(x) = 6x^2$.

   $f'(x) = 12x$. Factor out 6, apply power rule: $6 \cdot 2x^1$.

4. What is the derivative of $x^9$?

   $9x^8$. Power rule: $9 \cdot x^{9-1} = 9x^8$.

5. What is the derivative of $x^{\frac{9}{4}}$?

   $\frac{9}{4}x^{\frac{5}{4}}$. Power rule: $\frac{9}{4} \cdot x^{\frac{9}{4}-1} = \frac{9}{4}x^{\frac{5}{4}}$.

6. Differentiate: $f(x) = -3x^{4.5}$.

   $f'(x) = -13.5x^{3.5}$. Factor out $-3$, apply power rule: $-3 \cdot 4.5x^{3.5}$.

7. Differentiate: $f(x) = 2x^{-\frac{1}{4}}$.

   $f'(x) = -\frac{1}{2}x^{-\frac{5}{4}}$. Factor out 2, apply power rule: $2 \cdot (-\frac{1}{4})x^{-\frac{5}{4}}$.

8. What is the derivative of $x^{0.1}$?

   $0.1x^{-0.9}$. Power rule: $0.1 \cdot x^{0.1-1} = 0.1x^{-0.9}$.

9. What is the derivative of $x^{10}$?

   $10x^9$. Power rule: $10 \cdot x^{10-1} = 10x^9$.

10. Differentiate: $f(x) = x^{-10}$.

    $f'(x) = -10x^{-11}$. Power rule: $-10 \cdot x^{-10-1} = -10x^{-11}$.

11. What is the derivative of $x^{-5}$?

    $-5x^{-6}$. Power rule: $-5 \cdot x^{-5-1} = -5x^{-6}$.

12. What is the derivative of $x^{-7}$?

    $ -$7x^{-8}$ $. Power rule: $-7 \cdot x^{-7-1} = -7x^{-8}$.

13. What is the derivative of $x^{1.5}$?

    $1.5x^{0.5}$. Power rule: $1.5 \cdot x^{1.5-1} = 1.5x^{0.5}$.

14. State the derivative of $x^{\frac{3}{2}}$.

    $\frac{3}{2}x^{\frac{1}{2}}$. Power rule: $\frac{3}{2} \cdot x^{\frac{3}{2}-1} = \frac{3}{2}x^{\frac{1}{2}}$.

15. What is the derivative of $x^{\frac{7}{2}}$?

    $\frac{7}{2}x^{\frac{5}{2}}$. Power rule: $\frac{7}{2} \cdot x^{\frac{7}{2}-1} = \frac{7}{2}x^{\frac{5}{2}}$.

16. What is the derivative of $x^5$?

    $5x^4$. Power rule: $n \cdot x^{n-1} = 5 \cdot x^{5-1} = 5x^4$.

17. What is the derivative of $x^{\frac{1}{3}}$?

    $\frac{1}{3}x^{-\frac{2}{3}}$. Power rule: $\frac{1}{3} \cdot x^{\frac{1}{3}-1} = \frac{1}{3}x^{-\frac{2}{3}}$.

18. State the Power Rule for differentiation of $x^n$.

    $\frac{d}{dx}(x^n) = nx^{n-1}$. The fundamental rule for differentiating power functions.

19. What is the derivative of $x^{2.7}$?

    $2.7x^{1.7}$. Power rule: $2.7 \cdot x^{2.7-1} = 2.7x^{1.7}$.

20. Differentiate: $f(x) = x^{-\frac{1}{2}}$.

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

21. Differentiate: $f(x) = 8x^{3}$.

    $f'(x) = 24x^{2}$. Factor out 8, apply power rule: $8 \cdot 3x^2$.

22. What is the derivative of $x^{\frac{4}{3}}$?

    $\frac{4}{3}x^{\frac{1}{3}}$. Power rule: $\frac{4}{3} \cdot x^{\frac{4}{3}-1} = \frac{4}{3}x^{\frac{1}{3}}$.

23. Differentiate: $f(x) = -5x^{6}$.

    $f'(x) = -30x^{5}$. Factor out $-5$, apply power rule: $-5 \cdot 6x^5$.

24. Differentiate: $f(x) = 7x^4$.

    $f'(x) = 28x^3$. Factor out constant 7, then apply power rule: $7 \cdot 4x^3$.

25. Differentiate: $f(x) = x^3$.

    $f'(x) = 3x^2$. Apply power rule: bring down exponent 3, subtract 1 from exponent.

26. Differentiate: $f(x) = 7x^{0}$.

    $f'(x) = 0$. Since $7x^0 = 7$ (a constant), its derivative is 0.

27. Differentiate: $f(x) = -x^8$.

    $f'(x) = -8x^7$. Factor out $-1$, apply power rule: $-1 \cdot 8x^7$.

28. Differentiate: $f(x) = x^{-2}$.

    $f'(x) = -2x^{-3}$. Power rule with negative exponent: $-2 \cdot x^{-2-1}$.

29. Differentiate: $f(x) = 10x^{0.5}$.

    $f'(x) = 5x^{-0.5}$. Factor out 10, apply power rule: $10 \cdot 0.5x^{-0.5}$.

30. Differentiate: $f(x) = 9x^{-3}$.

    $f'(x) = -27x^{-4}$. Factor out constant 9, then apply power rule: $9 \cdot (-3)x^{-4}$.

31. What is the derivative of $x^{\frac{5}{3}}$?

    $\frac{5}{3}x^{\frac{2}{3}}$. Power rule: $\frac{5}{3} \cdot x^{\frac{5}{3}-1} = \frac{5}{3}x^{\frac{2}{3}}$.

32. Differentiate: $f(x) = -4x^6$.

    $f'(x) = -24x^5$. Factor out $-4$, apply power rule: $-4 \cdot 6x^5$.

33. Differentiate: $f(x) = 4x^{2.3}$.

    $f'(x) = 9.2x^{1.3}$. Factor out 4, apply power rule: $4 \cdot 2.3x^{1.3}$.

34. What is the derivative of $x^0$?

    $0$. Since $x^0 = 1$ (a constant), its derivative is 0.

35. What is the derivative of $x^1$?

    $1$. The derivative of $x$ (first power) is 1.

36. What is the derivative of $x^{-1}$?

    $-x^{-2}$. Power rule: $-1 \cdot x^{-1-1} = -x^{-2}$.

37. What is the derivative of a constant $c$?

    $0$. The derivative of any constant is always zero.

38. Differentiate: $f(x) = 3x^{\frac{2}{3}}$.

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

39. Differentiate: $f(x) = x^{\frac{4}{5}}$.

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

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