Sketching Graphs of Functions and Derivatives - AP Calculus AB

1. Differentiate $f(x) = \frac{1}{x}$.

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

2. State the Power Rule for differentiation.

   $\frac{d}{dx}x^n = nx^{n-1}$. Multiply by exponent, reduce exponent by 1.

3. Calculate $f'(x)$ for $f(x) = \text{cos}(x)$.

   $f'(x) = -\text{sin}(x)$. Derivative of cosine is negative sine.

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

   $f''(x) = 36x^2$. Apply power rule twice: $f'(x) = 12x^3$, then again.

5. Calculate $f'(x)$ for $f(x) = \text{arccos}(x)$.

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

6. Identify the critical points of $f(x) = x^3 - 3x$.

   $x = 0, x = \text{±}\frac{\text{√}3}{3}$. Set $f'(x) = 3x^2 - 3 = 0$ and solve for x.

7. Calculate $f'(x)$ for $f(x) = \frac{1}{2}x^4$.

   $f'(x) = 2x^3$. Apply power rule: $4 \cdot \frac{1}{2} \cdot x^3 = 2x^3$.

8. Find $f'(x)$ for $f(x) = \text{csc}(x)$.

   $f'(x) = -\text{csc}(x)\text{cot}(x)$. Derivative involves negative cotangent cosecant.

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

   $f'(x) = \frac{1}{\sqrt{1 - x^2}}$. Inverse trig derivative with radical denominator.

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

    $f'(x) = 2x$. Power rule: bring down exponent, subtract 1.

11. What is $f'(x)$ for $f(x) = \text{arctan}(x)$?

    $f'(x) = \frac{1}{1+x^2}$. Inverse tangent derivative with squared denominator.

12. What does $f'(x) = 0$ indicate about $f(x)$?

    Potential extrema. Horizontal tangent lines occur at critical points.

13. State the Quotient Rule for differentiation.

    $\frac{d}{dx}[\frac{u}{v}] = \frac{u'v - uv'}{v^2}$. Low d-high minus high d-low over low squared.

14. Find $f'(x)$ for $f(x) = \text{sec}(x)$.

    $f'(x) = \text{sec}(x)\text{tan}(x)$. Product of secant and tangent functions.

15. What is the Product Rule for differentiation?

    $\frac{d}{dx}[uv] = u'v + uv'$. Sum of each function times the other's derivative.

16. Identify the intervals where $f(x) = x^2 - 4$ is increasing.

    $x > 0$. $f'(x) = 2x > 0$ when $x > 0$.

17. State the Chain Rule for differentiation.

    $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$. Differentiate outer function times inner derivative.

18. Differentiate $f(x) = 7x^3 - 2x$.

    $f'(x) = 21x^2 - 2$. Apply power rule to each term separately.

19. What is $f'(x)$ for $f(x) = x^5$?

    $f'(x) = 5x^4$. Power rule: bring down 5, subtract 1 from exponent.

20. What indicates a local maximum in $f'(x)$?

    Change from positive to negative. Derivative sign changes from + to - at local max.

21. Which test identifies concavity?

    Second Derivative Test. Examines sign of $f''(x)$ to determine concavity.

22. What is $f'(x)$ for $f(x) = \text{sin}(x)$?

    $f'(x) = \text{cos}(x)$. Derivative of sine is cosine.

23. Find $f'(x)$ for $f(x) = \text{arccot}(x)$.

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

24. Which test uses $f''(x)$ to find extrema?

    Second Derivative Test. Uses second derivative to classify critical points.

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

    $f'(x) = -\text{csc}^2(x)$. Derivative of cotangent is negative cosecant squared.

26. What is the derivative of a constant $c$?

    $0$. Constants have zero rate of change.

27. Find the derivative of $f(x) = a^x$.

    $f'(x) = a^x \text{ln}(a)$. General exponential derivative includes $\ln(a)$.

28. Find $f'(x)$ for $f(x) = \text{ln}(x)$.

    $f'(x) = \frac{1}{x}$. The natural logarithm's derivative is $\frac{1}{x}$.

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

    $f'(x) = \frac{1}{\text{√}(1-x^2)}$. Inverse trig derivative with radical denominator.

30. Find the critical points of $f(x) = x^2 - 6x + 8$.

    $x = 3$. Set $f'(x) = 2x - 6 = 0$ and solve.

31. Identify the concavity of $f(x) = x^4$.

    Concave up for all $x$. $f''(x) = 12x^2 \geq 0$ for all real x.

32. What is the inverse function derivative formula?

    $[f^{-1}]'(x) = \frac{1}{f'(f^{-1}(x))}$. Reciprocal of derivative at corresponding point.

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

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

34. Calculate $f'(x)$ for $f(x) = \arccos(x)$.

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

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

    $f'(x) = \text{sec}^2(x)$. Derivative of tangent is secant squared.

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