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

1. What is the derivative of $cf(x)$, where $c$ is a constant?

   $(cf)' = cf'$. Constant multiple rule: factor out the constant.

2. Given $f(x) = x^2 + 3x$, find $f'(x)$.

   2x + 3. Apply power rule to each term separately.

3. State the sum rule: What is $\frac{d}{dx}[f(x)+g(x)]$?

   $f'(x)+g'(x)$. Differentiate each term separately and add.

4. What is $\frac{d}{dx}[3f(x)-2g(x)]$?

   $3f'(x)-2g'(x)$. Apply constant multiple rule to each term.

5. What is $\frac{d}{dx}[5f(x)]$?

   $5f'(x)$. Apply constant multiple rule: pull out 5.

6. What is $\frac{d}{dx}[-4f(x)+g(x)]$?

   $-4f'(x)+g'(x)$. Apply constant multiple rule to each term.

7. What is $\frac{d}{dx}[7-f(x)-g(x)]$?

   $-f'(x)-g'(x)$. Constant 7 has derivative 0; negative signs carry through.

8. Calculate the derivative of $5x - 3$.

   5. Derivative of linear function is the coefficient of $x$.

9. What is the derivative of a constant $c$ with respect to $x$?

   0. Constants have zero rate of change.

10. State the derivative rule for the sum of two functions $f(x)$ and $g(x)$.

    $(f + g)' = f' + g'$. Derivative of sum equals sum of derivatives.

11. What is $\frac{d}{dx}[-2g(x)]$?

    $-2g'(x)$. Apply constant multiple rule: pull out -2.

12. State the derivative of the function $-5$.

    0. Any constant has derivative zero.

13. What is $\frac{d}{dx}[\frac{1}{2}f(x)+\frac{3}{2}g(x)]$?

    $\frac{1}{2}f'(x)+\frac{3}{2}g'(x)$. Fractional constants follow the same rule.

14. State the constant rule: What is $\frac{d}{dx}[c]$ for a constant $c$?

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

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

    $0$. 7 is a constant, so its derivative is 0.

16. What is $\frac{d}{dx}[12-g(x)]$?

    $-g'(x)$. Derivative of 12 is 0; $-g(x)$ gives $-g'(x)$.

17. What is $\frac{d}{dx}[-3]$?

    $0$. -3 is a constant, so its derivative is 0.

18. State the difference rule: What is $\frac{d}{dx}[f(x)-g(x)]$?

    $f'(x)-g'(x)$. Differentiate each term separately and subtract.

19. What is $\frac{d}{dx}[f(x)+9]$?

    $f'(x)$. Derivative of constant 9 is 0, leaving $f'(x)$.

20. State the derivative rule for the difference of two functions $f(x)$ and $g(x)$.

    $(f - g)' = f' - g'$. Derivative of difference equals difference of derivatives.

21. What is $\frac{d}{dx}[2f(x)+3g(x)-5]$?

    $2f'(x)+3g'(x)$. Constant -5 has derivative 0; apply rules to rest.

22. State the derivative of $x^n$ where $n$ is a constant.

    $nx^{n-1}$. Power rule: bring down exponent, reduce by 1.

23. What is $\frac{d}{dx}[f(x)-g(x)+h(x)]$?

    $f'(x)-g'(x)+h'(x)$. Apply sum/difference rules term by term.

24. Find the derivative of $8x - 2$.

    8. Derivative of linear term is its coefficient.

25. What is $\frac{d}{dx}[f(x)+g(x)+h(x)]$?

    $f'(x)+g'(x)+h'(x)$. Sum rule extends to any number of functions.

26. Calculate the derivative of $12x$.

    12. Derivative of linear term $12x$ is the coefficient.

27. Find and correct the derivative: $\frac{d}{dx}[f(x)+g(x)]=f'(x)g'(x)$.

    Correct: $\frac{d}{dx}[f(x)+g(x)]=f'(x)+g'(x)$. Sum rule gives addition, not multiplication of derivatives.

28. What is $\frac{d}{dx}[-\frac{5}{3}f(x)-\frac{1}{3}]$?

    $-\frac{5}{3}f'(x)$. Constant $-\frac{1}{3}$ has derivative 0.

29. Calculate the derivative of $7x^3 - 4x$.

    $21x^2$ - 4. Use power rule: $7 \cdot 3x^2 - 4 = 21x^2 - 4$.

30. State the constant multiple rule: What is $\frac{d}{dx}[c\text{ }f(x)]$ for constant $c$?

    $c\text{ }f'(x)$. Pull out the constant and multiply by the derivative.

31. Find and correct the derivative: $\frac{d}{dx}[3f(x)]=3f(x)$.

    Correct: $\frac{d}{dx}[3f(x)]=3f'(x)$. Must differentiate $f(x)$, not just copy it.

32. What is the derivative of a constant multiplied by $x$, $7x$?

    7. Constant multiple rule: derivative of $cx$ is $c$.

Study these flashcards