Selecting Procedures for Calculating Derivatives - AP Calculus BC

1. Determine the derivative of $f(x) = x^3 - 4x + 6$.

   $f'(x) = 3x^2 - 4$. Apply power rule term by term.

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

   $f'(x) = 2x\text{sec}^2(x^2)$. Chain rule with tangent: $\sec^2(x^2) \cdot 2x$.

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

   $f'(x) = 2x\text{e}^{x^2}$. Chain rule: $e^{x^2} \cdot 2x$.

4. What is the derivative of $\text{sec}(x)$ with respect to $x$?

   $\frac{d}{dx}[\text{sec}(x)] = \text{sec}(x)\text{tan}(x)$. Secant derivative involves tangent.

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

   $f'(x) = -2x\text{sin}(x^2)$. Chain rule: $-\sin(x^2) \cdot 2x$.

6. Find the derivative of $f(x) = 3x^4 + 2x^2 - 5$.

   $f'(x) = 12x^3 + 4x$. Apply power rule to each term separately.

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

   $f'(x) = 4x^3 - 6x^{-3}$. Power rule applied to positive and negative exponents.

8. Identify the product rule for derivatives.

   $(uv)' = u'v + uv'$. Product rule: first times derivative of second plus second times derivative of first.

9. Calculate the derivative of $f(x) = \text{sin}(x) + \text{cos}(x)$.

   $f'(x) = \text{cos}(x) - \text{sin}(x)$. Sum rule: derivative of sum equals sum of derivatives.

10. State the formula for the derivative of $\text{ln}(x)$.

    $\frac{d}{dx}[\text{ln}(x)] = \frac{1}{x}$. Natural log derivative is reciprocal function.

11. State the formula for the derivative of $\text{sin}(x)$.

    $\frac{d}{dx}[\text{sin}(x)] = \text{cos}(x)$. Derivative of sine is cosine.

12. Calculate the derivative of $f(x) = \text{cos}^3(x)$.

    $f'(x) = -3\text{cos}^2(x)\text{sin}(x)$. Chain rule: $3\cos^2(x) \cdot (-\sin(x))$.

13. What is the derivative of $\text{cos}(x)$ with respect to $x$?

    $\frac{d}{dx}[\text{cos}(x)] = -\text{sin}(x)$. Derivative of cosine is negative sine.

14. What is the derivative of $\text{cot}(x)$ with respect to $x$?

    $\frac{d}{dx}[\text{cot}(x)] = -\text{csc}^2(x)$. Cotangent derivative is negative cosecant squared.

15. Identify the derivative of $f(x) = \text{csc}(x^3)$.

    $f'(x) = -3x^2\text{csc}(x^3)\text{cot}(x^3)$. Chain rule with cosecant function.

16. What is the derivative of $f(x) = \text{sec}(2x)$?

    $f'(x) = 2\text{sec}(2x)\text{tan}(2x)$. Chain rule with secant function.

17. What is the derivative of $f(x) = x^n$ with respect to $x$?

    $f'(x) = nx^{n-1}$. Power rule: bring down exponent, reduce power by 1.

18. What is the derivative of $e^x$ with respect to $x$?

    $\frac{d}{dx}[e^x] = e^x$. Exponential function is its own derivative.

19. Find the derivative of $f(x) = e^{2x}$ with respect to $x$.

    $f'(x) = 2e^{2x}$. Chain rule: derivative of inside times $e^{2x}$.

20. Identify the chain rule for derivatives.

    If $y = f(g(x))$, then $y' = f'(g(x))g'(x)$. Chain rule for composite functions.

21. Compute the derivative of $f(x) = x \text{ln}(x)$.

    $f'(x) = \text{ln}(x) + 1$. Product rule: $1 \cdot \ln(x) + x \cdot \frac{1}{x}$.

22. State the power rule for derivatives.

    If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Multiply by exponent, reduce power by 1.

23. Compute the derivative of $f(x) = x \ln(x)$.

    $f'(x) = \ln(x) + 1$. Product rule: $1 \cdot \ln(x) + x \cdot \frac{1}{x}$.

24. Determine the derivative of $f(x) = \frac{\text{e}^x}{x}$.

    $f'(x) = \frac{\text{e}^x(x-1)}{x^2}$. Quotient rule with $u = e^x, v = x$.

25. Find the derivative of $f(x) = \text{sec}^2(x)$.

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

26. What is the derivative of $f(x) = \text{ln}(x^2 + 1)$?

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

27. Calculate the derivative of $f(x) = \text{tan}(3x)$.

    $f'(x) = 3\text{sec}^2(3x)$. Chain rule with tangent: $\sec^2(3x) \cdot 3$.

28. What is the derivative of $f(x) = \text{e}^{\text{sin}(x)}$?

    $f'(x) = \text{e}^{\text{sin}(x)}\text{cos}(x)$. Chain rule: $e^{\sin(x)} \cdot \cos(x)$.

29. State the quotient rule for derivatives.

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

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

    $f'(x) = \frac{\text{cos}(x)}{\text{sin}(x)}$. Chain rule: $\frac{1}{\sin(x)} \cdot \cos(x)$.

31. Calculate the derivative of $f(x) = 4x^5 - 2x^3 + x$.

    $f'(x) = 20x^4 - 6x^2 + 1$. Power rule applied to polynomial.

32. Compute the derivative of $f(x) = \frac{1}{x}$.

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

33. Compute the derivative of $f(x) = \frac{1}{x^2 + 1}$.

    $f'(x) = -\frac{2x}{(x^2 + 1)^2}$. Quotient rule with $u = 1, v = x^2 + 1$.

34. Calculate the derivative of $f(x) = \sin(x) + \cos(x)$.

    $f'(x) = \cos(x) - \sin(x)$. Sum rule: derivative of sum equals sum of derivatives.

35. State the derivative of $\text{tan}(x)$ with respect to $x$.

    $\frac{d}{dx}[\text{tan}(x)] = \text{sec}^2(x)$. Derivative of tangent is secant squared.

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

    $f'(x) = 2\text{sin}(x)\text{cos}(x)$. Chain rule: $2\sin(x)\cos(x)$.

37. What is the derivative of $\text{csc}(x)$ with respect to $x$?

    $\frac{d}{dx}[\text{csc}(x)] = -\text{csc}(x)\text{cot}(x)$. Cosecant derivative involves cotangent.

38. Identify the derivative of $f(x) = x^{5/3}$.

    $f'(x) = \frac{5}{3}x^{2/3}$. Power rule with fractional exponent.

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

    $f'(x) = \frac{1}{x}$. Chain rule: $\frac{d}{dx}[\ln(2x)] = \frac{1}{2x} \cdot 2 = \frac{1}{x}$.

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

    $f'(x) = 2x\text{sin}(x) + x^2\text{cos}(x)$. Product rule: $u'v + uv'$ where $u = x^2, v = \sin(x)$.

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