Derivative Rules of Constant, Sum, Difference - AP Calculus BC

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

   $-3x^2$. Constant term has zero derivative.

2. State the constant multiple rule for derivatives.

   $\frac{d}{dx}[cf(x)] = c \frac{d}{dx}[f(x)]$. Factor out the constant when differentiating.

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

   2x - 4. Differentiate each term, constant disappears.

4. What is the derivative of $f(x) = 5x + 6$?

   5. Derivative of $5x$ is $5$, constant disappears.

5. State the derivative of $f(x) = 2x^2$ using the constant multiple rule.

   4x. Apply constant multiple rule: $2 \cdot 2x = 4x$.

6. State the derivative of $f(x) = 12$.

   $0$. Any constant has zero derivative.

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

   2x - 2. Apply difference rule: $2x - 2$.

8. What is the derivative of a constant function $f(x) = c$?

   0. Constants have zero rate of change.

9. State the difference rule for derivatives.

   $\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$. Derivative of difference equals difference of derivatives.

10. What is the derivative of $f(x) = 7$?

    0. Any constant has zero derivative.

11. What is the derivative of $f(x) = 4x - 4$?

    4. Derivative of $4x$ is $4$, constant disappears.

12. What is the derivative of $f(x) = 8x^2 + 3$?

    16x. Derivative of $8x^2$ is $16x$, constant term is zero.

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

    9. Derivative of $x$ is $1$, multiply by $9$.

14. Find $\frac{d}{dx}[2x^2 + 5x + 1]$.

    $4x + 5$. Differentiate each term separately.

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

    2 + 6x. Apply sum rule to each term.

16. What is $\frac{d}{dx}[7]$?

    0. Any constant has zero derivative.

17. Find the derivative of $f(x) = x^3 + x^2 + x$.

    $3x^2$ + 2x + 1. Apply power rule to each term.

18. State the derivative of $f(x) = 5x^2 + 4x$.

    10x + 4. Apply power rule to each term.

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

    4x - 3. Differentiate each term, constant disappears.

20. What is the derivative of $f(x) = x^4 + 2x^3$?

    $4x^3$ + $6x^2$. Apply power rule: $4x^3 + 6x^2$.

21. State the rule for $\frac{d}{dx}[c]$ where $c$ is a constant.

    0. Constants have zero rate of change.

22. What is the result of $\frac{d}{dx}[4x^3]$?

    $12x^2$. Apply constant multiple rule: $4 \cdot 3x^2 = 12x^2$.

23. Find the derivative of $f(x) = 6$.

    0. Any constant has derivative zero.

24. Find the derivative of $f(x) = 9x^2 - 2x + 4$.

    18x - 2. Apply power rule to each term.

25. Find $\frac{d}{dx}[3x^3 - 4x + 5]$.

    $9x^2$ - 4. Differentiate each term using power rule.

26. State the derivative of $f(x) = x^3 - 3x$.

    $3x^2 - 3$. Apply power rule to each term.

27. What is the derivative of $f(x) = 5x^4$?

    $20x^3$. Apply constant multiple rule: $5 \cdot 4x^3 = 20x^3$.

28. What is the derivative of $f(x) = 6x^2 + 7$?

    12x. Derivative of $6x^2$ is $12x$, constant disappears.

29. Find the derivative of $f(x) = 7x^3$.

    $21x^2$. Apply constant multiple rule: $7 \cdot 3x^2 = 21x^2$.

30. State the sum rule for derivatives.

    $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$. Derivative of sum equals sum of derivatives.

31. State the derivative of $f(x) = 0$.

    0. Zero function has zero derivative.

32. What is the derivative of $f(x) = 3x^3 + 4x$?

    $9x^2$ + 4. Apply sum rule: $9x^2 + 4$.

33. Find the derivative of $f(x) = 10$.

    0. Any constant has zero derivative.

34. What is the derivative of $f(x) = x - 5$?

    1. Derivative of $x$ is $1$, constant disappears.

35. Find the derivative of $f(x) = 5x^2$ using the constant multiple rule.

    10x. Apply constant multiple rule: $5 \cdot 2x = 10x$.

36. State the derivative of $f(x) = 3x^2 + 2x$.

    6x + 2. Apply power rule to each term.

37. What is the derivative of $f(x) = 4x^3 - 9$?

    $12x^2$. Derivative of $4x^3$ is $12x^2$, constant disappears.

38. Find the derivative of $f(x) = 4x^2 - 7x$.

    8x - 7. Apply power rule to each term.

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