Rates of Change - AP Precalculus

1. What is the derivative of $f(x) = \log_a(x)$?

   $f'(x) = \frac{1}{x \ln(a)}$. Change of base gives factor $\frac{1}{\ln(a)}$.

2. Identify the derivative of $f(x) = e^x$.

   $f'(x) = e^x$. Exponential function $e^x$ is its own derivative.

3. What is the derivative of $f(x) = \text{arccsc}(x)$?

   $f'(x) = -\frac{1}{|x|\text{sqrt}(x^2-1)}$. Negative of arcsecant derivative.

4. What is the rate of change of $f(x) = 7$?

   Rate of change is 0. Constant functions have zero rate of change.

5. What is the rate of change of $f(x) = x^2 + 3x$ at $x=1$?

   Rate of change is 5. Use $f'(x) = 2x + 3$, evaluate at $x = 1$.

6. What is the derivative of $f(x) = \frac{1}{x^3}$?

   $f'(x) = -\frac{3}{x^4}$. Rewrite as $x^{-3}$ and apply power rule.

7. What is the derivative of $f(x) = \text{ln}(x^2)$?

   $f'(x) = \frac{2}{x}$. Use chain rule: $\frac{1}{x^2} \cdot 2x = \frac{2}{x}$.

8. Identify the derivative of $f(x) = \text{tan}(x)$.

   $f'(x) = \text{sec}^2(x)$. Tangent differentiates to secant squared.

9. What is the derivative of $f(x) = \text{log}_a(x)$?

   $f'(x) = \frac{1}{x\text{ln}(a)}$. Change of base gives factor $\frac{1}{\ln(a)}$.

10. Find $f'(x)$ for $f(x) = \text{sin}(x)$.

    $f'(x) = \text{cos}(x)$. Sine differentiates to cosine.

11. What is the derivative of $f(x) = \text{arccos}(x)$?

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

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

    $f'(x) = x^x(\text{ln}(x)+1)$. Use logarithmic differentiation for variable base and exponent.

13. State the constant rule for derivatives.

    Derivative of a constant is 0. Constants have zero slope everywhere.

14. Determine the derivative of $f(x) = 5^x$.

    $f'(x) = 5^x \ln(5)$. General formula $a^x$ derivative includes $\ln(a)$ factor.

15. What is the derivative of $f(x) = \text{cos}(x)$?

    $f'(x) = -\text{sin}(x)$. Cosine differentiates to negative sine.

16. Calculate $\frac{d}{dx}[x^5 - 2x^3]$.

    $5x^4 - 6x^2$. Apply power rule to each term separately.

17. If $y = 3x^3 - 5x$, what is $\frac{dy}{dx}$?

    $\frac{dy}{dx} = 9x^2 - 5$. Use power rule: $3 \cdot 3x^2 - 5 \cdot 1$.

18. Calculate $\frac{d}{dx}[5x^2]$.

    $10x$. Factor out constant 5, apply power rule.

19. Define average rate of change of a function $f(x)$ over $[a, b]$.

    $\frac{f(b) - f(a)}{b - a}$. Slope formula between two points on the function.

20. What is the rate of change of $f(x) = x^4$ at $x = 2$?

    Rate of change is 32. Use $f'(x) = 4x^3$, evaluate at $x = 2$.

21. What is the instantaneous rate of change at $x = a$?

    The derivative $f'(a)$. Limit of average rate of change as interval approaches zero.

22. State the derivative of $f(x) = \text{sin}^2(x)$.

    $f'(x) = 2\text{sin}(x)\text{cos}(x)$. Use chain rule on $\sin^2(x)$.

23. Determine the derivative of $f(x) = \text{cot}(x)$.

    $f'(x) = -\text{csc}^2(x)$. Cotangent is $\frac{\cos(x)}{\sin(x)}$, use quotient rule.

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

    $f'(x) = 2x$. Apply power rule: bring down exponent 2, reduce power by 1.

25. State the derivative of $f(x) = \text{arcsin}(x)$.

    $f'(x) = \frac{1}{\text{sqrt}(1-x^2)}$. Standard inverse trig derivative formula.

26. State the power rule for derivatives.

    $\frac{d}{dx}[x^n] = nx^{n-1}$. Multiply by exponent, then decrease exponent by 1.

27. Calculate the derivative of $f(x) = \text{e}^{2x}$.

    $f'(x) = 2\text{e}^{2x}$. Chain rule: multiply by inner derivative 2.

28. Identify the derivative of $f(x) = \text{arctan}(x)$.

    $f'(x) = \frac{1}{1+x^2}$. Standard inverse trig derivative for arctangent.

29. Find the rate of change of $f(x) = 4x - 7$ at $x = 2$.

    Rate of change is 4. Linear function has constant slope 4.

30. Find the derivative of $f(x) = \text{sec}(x)$.

    $f'(x) = \text{sec}(x)\text{tan}(x)$. Product rule applied to $\sec(x) = \frac{1}{\cos(x)}$.

31. Determine $f'(x)$ for $f(x) = \text{ln}(x)$.

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

32. Determine the derivative of $f(x) = \text{arcsec}(x)$.

    $f'(x) = \frac{1}{|x|\text{sqrt}(x^2-1)}$. Involves absolute value for proper domain.

33. Identify the derivative of $f(x) = \text{tan}^2(x)$.

    $f'(x) = 2\text{tan}(x)\text{sec}^2(x)$. Chain rule applied to $\tan^2(x)$.

34. Determine the derivative of $f(x) = x^{\frac{1}{2}}$.

    $f'(x) = \frac{1}{2}x^{-\frac{1}{2}}$. Power rule with fractional exponent $\frac{1}{2}$.

35. Define instantaneous rate of change.

    The derivative at a specific point. The slope of tangent line at one point.

36. Identify the derivative of $f(x) = \tan(x)$.

    $f'(x) = \sec^2(x)$. Tangent differentiates to secant squared.

37. Calculate the derivative of $f(x) = \text{arccot}(x)$.

    $f'(x) = -\frac{1}{1+x^2}$. Negative of arctangent derivative.

38. What is the rate of change for $f(x) = x^3$ at $x = 1$?

    Rate of change is 3. Use $f'(x) = 3x^2$, evaluate at $x = 1$.

39. What is the derivative of $f(x) = \text{cos}^2(x)$?

    $f'(x) = -2\text{sin}(x)\text{cos}(x)$. Chain rule gives negative of $\sin^2(x)$ derivative.

40. What is the derivative of $f(x) = \frac{1}{x}$?

    $f'(x) = -\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply power rule.

41. What is the derivative of $f(x) = \text{csc}(x)$?

    $f'(x) = -\text{csc}(x)\text{cot}(x)$. Quotient rule applied to $\csc(x) = \frac{1}{\sin(x)}$.

Study these flashcards