Rates of Change in Applied Concepts - AP Calculus BC

1. Find the rate of change of the volume of a sphere with respect to its radius.

   $\frac{dV}{dr} = 4\pi r^2$. Derivative of $V = \frac{4}{3}\pi r^3$ using power rule.

2. Evaluate the derivative of $f(x) = \frac{1}{x}$ at $x = 1$.

   $-\frac{1}{x^2}$, so $f'(1) = -1$. Reciprocal function derivative using power rule.

3. Determine the rate of change for $f(x) = x^2$ at $x = 5$.

   $f'(x) = 2x$, so $f'(5) = 10$. Power rule gives $2x$, substitute $x = 5$.

4. What is the derivative of $f(x) = a^x$ where $a$ is a constant?

   $f'(x) = a^x \ln(a)$. Exponential derivative includes natural log factor.

5. Evaluate $\frac{d}{dx}(x^4)$ at $x = 1$.

   $4x^3$, so $f'(1) = 4$. Power rule gives $4x^3$, substitute $x = 1$.

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

   $f'(x) = 0$. Derivative of constant is always zero.

7. Determine the derivative of $f(x) = \cos(x)$.

   $f'(x) = -\sin(x)$. Cosine derivative is negative sine.

8. State the product rule for the derivative of $u(x)v(x)$.

   $(uv)' = u'v + uv'$. Product rule for multiplied functions.

9. Determine the derivative of $f(x) = \arctan(x)$.

   $f'(x) = \frac{1}{1+x^2}$. Arctangent derivative formula.

10. Identify the derivative of $f(x) = x^n$ using the power rule.

    $f'(x) = nx^{n-1}$. Standard power rule for polynomial derivatives.

11. Determine $\frac{d}{dx}(x^2 + 3x + 2)$.

    $2x + 3$. Sum rule and power rule applied term by term.

12. What is the derivative of $f(x) = \cot(x)$?

    $f'(x) = -\csc^2(x)$. Cotangent derivative is negative cosecant squared.

13. Find the derivative of $f(x) = \arcsin(x)$.

    $f'(x) = \frac{1}{\sqrt{1-x^2}}$. Arcsine derivative formula.

14. Calculate the derivative of $f(x) = \ln(ax)$ where $a$ is a constant.

    $f'(x) = \frac{1}{x}$. Constant factor $a$ cancels in logarithm derivative.

15. Calculate the derivative of $f(x) = 2x^3 - 3x^2 + x$.

    $6x^2 - 6x + 1$. Power rule applied to polynomial terms.

16. Calculate the rate of change of $f(x) = x^2 + 2x$ at $x = 1$.

    $f'(x) = 2x + 2$, so $f'(1) = 4$. Apply power and sum rules, then evaluate.

17. Find the derivative of $f(x) = \sec(x)$.

    $f'(x) = \sec(x)\tan(x)$. Secant derivative formula.

18. Calculate the derivative of $f(x) = \csc(x)$.

    $f'(x) = -\csc(x)\cot(x)$. Cosecant derivative formula.

19. What is the derivative of $f(x) = \arccos(x)$?

    $f'(x) = -\frac{1}{\sqrt{1-x^2}}$. Arccosine derivative is negative of arcsine.

20. Identify the rate of change of $y = x^3$ at $x = 2$.

    $f'(x) = 3x^2$, so $f'(2) = 12$. Apply power rule then substitute $x = 2$.

21. Calculate the derivative of $f(x) = 5x^2 - 3x + 7$.

    $10x - 3$. Power rule applied to polynomial.

22. Determine $\frac{d}{dx}(7x^2 - 4x + 1)$.

    $14x - 4$. Apply power rule to each term.

23. State the Chain Rule formula for derivatives.

    $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Composite function differentiation rule.

24. Find the derivative of $f(x) = e^x$.

    $f'(x) = e^x$. Exponential function derivative equals itself.

25. What is the derivative of $f(x) = \ln(x)$?

    $f'(x) = \frac{1}{x}$. Natural logarithm derivative is reciprocal.

26. What is the rate of change of $y = 2x^3$ at $x = 2$?

    $f'(x) = 6x^2$, so $f'(2) = 24$. Constant multiple rule with power rule.

27. Identify the quotient rule for $\frac{u(x)}{v(x)}$.

    $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$. Quotient rule for divided functions.

28. Determine $\frac{d}{dx}(5x - 1)$ at $x = 1$.

    $5$, so the rate is $5$. Linear function has constant derivative.

29. Evaluate $\frac{d}{dx}(x^2 + x)$ at $x = 2$.

    $2x + 1$, so $f'(2) = 5$. Sum rule applied, then evaluate at $x = 2$.

30. What is the derivative of $f(x) = \tan(x)$?

    $f'(x) = \sec^2(x)$. Tangent derivative is secant squared.

31. Find the rate of change of $f(x) = x^3$ at $x = 4$.

    $f'(x) = 3x^2$, so $f'(4) = 48$. Power rule gives $3x^2$, substitute $x = 4$.

32. What is the formula for the rate of change of a function $f(x)$?

    $f'(x)$. The derivative represents instantaneous rate of change.

33. Find $\frac{d}{dx} (3x^3 - 5x^2 + 4)$.

    $9x^2 - 10x$. Power rule applied to each term.

34. What is the rate of change of the area of a circle with respect to its radius?

    $\frac{dA}{dr} = 2\pi r$. Derivative of $A = \pi r^2$ using power rule.

35. Find the rate of change for $f(x) = 4x^2$ at $x = 3$.

    $f'(x) = 8x$, so $f'(3) = 24$. Power rule gives $8x$, substitute $x = 3$.

36. Calculate the derivative of $f(x) = x^{-1}$.

    $f'(x) = -x^{-2}$. Power rule applied to negative exponent.

37. Calculate the derivative of $f(x) = \sin(x)$.

    $f'(x) = \cos(x)$. Sine derivative is cosine.

38. Evaluate $\frac{d}{dx}(x^3 - 2x + 4)$ at $x = 0$.

    $3x^2 - 2$, so $f'(0) = -2$. Differentiate then evaluate at $x = 0$.

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