Rates of Change in Applied Concepts - AP Calculus AB

1. What is the derivative of the function $f(x) = x^5 - 4x^3 + 2$?

   $5x^4 - 12x^2$. Power rule: $5x^4$ from $x^5$ and $-12x^2$ from $-4x^3$.

2. Determine the derivative of the function $y = \frac{2}{x^2}$.

   $-\frac{4}{x^3}$. Power rule: $2x^{-2}$ becomes $-4x^{-3}$.

3. Determine the derivative of $f(x) = \frac{3}{x}$.

   $-\frac{3}{x^2}$. Constant multiple rule: $3 \cdot (-x^{-2}) = -3x^{-2}$.

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

   $4x^3 + 4x$. Power rule: $4x^3$ from $x^4$ and $4x$ from $2x^2$.

5. Find the derivative of $y = 2x^3 + 3x^2 - 5x + 1$.

   $6x^2 + 6x - 5$. Power rule applied: $6x^2$, $6x$, and $-5$ from each term.

6. Identify the rate of change of the volume of a rectangular prism with respect to its height.

   Area of base. Volume formula is base area times height.

7. Find the derivative of $h(x) = 7x^5 - 3x^2 + 1$.

   $35x^4 - 6x$. Power rule: $35x^4$ from $7x^5$ and $-6x$ from $-3x^2$.

8. What is the derivative of the function $f(x) = \frac{2}{x^3}$?

   $-\frac{6}{x^4}$. Power rule: $2x^{-3}$ becomes $-6x^{-4}$.

9. What is the rate of change of the surface area of a sphere with respect to its radius?

   $8\pi r$. Derivative of sphere surface area $4\pi r^2$.

10. Identify the derivative of $y = 6x^2 + 4x - 5$.

    $12x + 4$. Power rule: $12x$ from $6x^2$ and $4$ from $4x$.

11. What is the derivative of the function $h(x) = \frac{2}{x^3}$?

    $-\frac{6}{x^4}$. Power rule: $2x^{-3}$ becomes $-6x^{-4}$.

12. What is the derivative of $y = x^3 + x^2$ with respect to $x$?

    $3x^2 + 2x$. Apply power rule: $3x^2$ from $x^3$ and $2x$ from $x^2$.

13. Find the rate of change of the circumference of a circle with respect to its radius.

    $2\pi$. Derivative of circumference $2\pi r$ is constant $2\pi$.

14. What is the rate of change of the perimeter of a square with respect to its side length?

    4. Each side contributes 1 to perimeter, so total rate is 4.

15. Find the rate of change of the volume of a cylinder with respect to its height.

    $\pi r^2$. For cylinder volume $\pi r^2 h$, derivative with respect to height.

16. What is the derivative of the function $g(x) = \frac{1}{x^4}$?

    $-\frac{4}{x^5}$. Power rule: $x^{-4}$ becomes $-4x^{-5}$.

17. Identify the derivative of the function $g(x) = \frac{1}{x}$.

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

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

    $3x^2 - 2$. Power rule: $3x^2$ from $x^3$ and $-2$ from $-2x$.

19. Determine the derivative of $f(x) = 4x^2 - 3x + 7$.

    $8x - 3$. Power rule: $8x$ from $4x^2$ and $-3$ from $-3x$.

20. What is the rate of change of the area of a rectangle with respect to its length?

    Width. For rectangle area $lw$, derivative with respect to length is width.

21. What is the derivative of the area of a circle with respect to its radius?

    $2\pi r$. Derivative of area $\pi r^2$ gives the circumference formula.

22. What is the derivative of $h(x) = 5x^3 - 4x + 8$?

    $15x^2 - 4$. Power rule: $15x^2$ from $5x^3$ and $-4$ from $-4x$.

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

    $6x^2 - 2x$. Power rule: $6x^2$ from $2x^3$ and $-2x$ from $-x^2$.

24. Determine the derivative of $g(x) = x^2 - \frac{1}{x}$.

    $2x + \frac{1}{x^2}$. Sum rule: $2x$ from $x^2$ plus $x^{-2}$ from $-x^{-1}$.

25. Identify the derivative of $y = 3x^3 + 5x^2 - x$.

    $9x^2 + 10x - 1$. Power rule: $9x^2$ from $3x^3$, $10x$ from $5x^2$, $-1$ from $-x$.

26. What is the rate of change of the surface area of a cube with respect to its side length?

    $12s$. Derivative of surface area $6s^2$ with respect to side.

27. What is the derivative of $h(x) = \frac{1}{x^5}$?

    $-\frac{5}{x^6}$. Power rule: $x^{-5}$ becomes $-5x^{-6}$.

28. What is the derivative of the function $g(x) = x^4 - \frac{1}{x^2}$?

    $4x^3 + \frac{2}{x^3}$. Power rule: $4x^3$ from $x^4$ and $2x^{-3}$ from $-x^{-2}$.

29. Find the rate of change of the area of a square with side length $s$.

    $2s$. Derivative of area $s^2$ gives rate of change.

30. Identify the rate of change of the area of a rectangle with respect to its width.

    Length. For rectangle area $lw$, derivative with respect to width is length.

31. What is the formula for the rate of change of the volume of a sphere with respect to its radius?

    $4\pi r^2$. Derivative of sphere volume $\frac{4}{3}\pi r^3$.

32. What is the formula for the derivative of $y = \frac{1}{x^3}$?

    $-\frac{3}{x^4}$. Power rule: $x^{-3}$ becomes $-3x^{-4}$

33. What is the derivative of $g(x) = x^3 + \frac{1}{x}$?

    $3x^2 - \frac{1}{x^2}$. Power rule: $3x^2$ from $x^3$ and $-x^{-2}$ from $x^{-1}$.

34. Identify the derivative of $y = \frac{1}{x} + x^2$.

    $-\frac{1}{x^2} + 2x$. Combine derivatives: $ -$x^{-2}$ $ and $ 2x $ terms.

35. What is the derivative of $h(x) = \frac{1}{x^2}$?

    $-\frac{2}{x^3}$. Power rule: $x^{-2}$ becomes $-2x^{-3}$.

36. Identify the rate of change of the volume of a cube with respect to its side length.

    $3s^2$. Derivative of volume $s^3$ with respect to side length.

37. Determine the derivative of the function $f(x) = 5x^4 - 3x^2 + 7$.

    $20x^3 - 6x$. Power rule applied: $20x^3$ from $5x^4$ and $-6x$ from $-3x^2$.

38. Find the derivative of $y = 4x^4 - 3x^3 + x - 7$.

    $16x^3 - 9x^2 + 1$. Power rule applied to each term: $16x^3$, $-9x^2$, $1$.

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