The Product Rule - AP Calculus AB

1. Find $d/dx$ of $f(x) = x^2 \times \text{sec}(x)$. Use Product Rule.

   $2x \times \text{sec}(x) + x^2 \times \text{sec}(x)\text{tan}(x)$. Product Rule: $(x^2)' \times \sec(x) + x^2 \times (\sec(x))'$.

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

   $2x \times \text{sin}(x) + x^2 \times \text{cos}(x)$. Product Rule: $(x^2)' \times \sin(x) + x^2 \times (\sin(x))'$.

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

   $3x^2 - 2x + 2$. Product Rule: $1 \times (x^2 - x + 1) + (x + 3) \times (2x - 1)$.

4. Which rule differentiates the product of two differentiable functions?

   The Product Rule. Standard rule for finding derivatives of function products.

5. Find $d/dx$ of $f(x) = x \times \text{cos}(x)$ using Product Rule.

   $-\text{sin}(x) + x \times -\text{sin}(x)$. Product Rule: $1 \times \cos(x) + x \times (-\sin(x))$.

6. Is the Product Rule applicable to $f(x) = (x^2 + x)(x - 1)$?

   Yes, it is applicable. Two differentiable functions multiplied together require Product Rule.

7. State the Product Rule formula for differentiation.

   $(f \times g)' = f' \times g + f \times g'$. Standard formula where each function multiplies the other's derivative.

8. What is the outcome of differentiating $f(x) = xe^x$ using Product Rule?

   $e^x + xe^x$. Product Rule: $1 \times e^x + x \times e^x$.

9. Calculate the derivative of $h(x) = \text{ln}(x) \times \text{tan}(x)$.

   $\frac{1}{x} \times \text{tan}(x) + \text{ln}(x) \times \text{sec}^2(x)$. Product Rule: $(\ln(x))' \times \tan(x) + \ln(x) \times (\tan(x))'$.

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

    $3x^2 + 4x + 2$. Product Rule: $1 \times (x^2 + 1) + (x + 2) \times 2x$.

11. Identify the derivative: $(3x^2) \times (2x^3)$. Use Product Rule.

    $12x^4 + 18x^5$. Apply Product Rule: $f' = 6x, g' = 6x^2$, then $f'g + fg'$.

12. Calculate the derivative: $y = (x^4)(\text{sin}(x))$ using Product Rule.

    $4x^3 \text{sin}(x) + x^4 \text{cos}(x)$. Product Rule: $(x^4)' \times \sin(x) + x^4 \times (\sin(x))'$.

13. What is $d/dx$ of $f(x) = (x^2)(\text{cos}(x))$ using Product Rule?

    $2x \text{cos}(x) - x^2 \text{sin}(x)$. Product Rule: $(x^2)' \times \cos(x) + x^2 \times (\cos(x))'$.

14. What do $f(x)$ and $g(x)$ represent in the Product Rule?

    Differentiable functions of $x$. They represent any two functions that can be differentiated.

15. Does the Product Rule apply to $f(x) = x^2 \times x^3$?

    Yes, but it can be simplified before applying. Could simplify to $x^5$ first, but Product Rule still applies.

16. Which component is found first in the Product Rule: $f'(x)$ or $g'(x)$?

    Either order is acceptable. The Product Rule is symmetric; order doesn't matter.

17. Calculate the derivative of $y = (2x^2)(\text{cos}(x))$ using Product Rule.

    $4x \text{cos}(x) - 2x^2 \text{sin}(x)$. Product Rule: $(2x^2)' \times \cos(x) + 2x^2 \times (\cos(x))'$.

18. Find the derivative: $f(x) = (2x + 1)(x^2 - 4x + 4)$. Use Product Rule.

    $2x^3 - 6x^2 + 10x - 4$. Product Rule: $2 \times (x^2 - 4x + 4) + (2x + 1) \times (2x - 4)$.

19. What is the result of differentiating $u(x) = e^x \times \text{ln}(x)$?

    $e^x \times \text{ln}(x) + \frac{e^x}{x}$. Product Rule: $(e^x)' \times \ln(x) + e^x \times (\ln(x))'$.

20. Find the derivative of $h(x) = x^2 \times \text{ln}(x)$.

    $2x \text{ln}(x) + x$. Product Rule: $(x^2)' \times \ln(x) + x^2 \times (\ln(x))'$.

21. Determine the derivative: $y = (x^5)(\text{cos}(x))$. Use Product Rule.

    $5x^4 \text{cos}(x) - x^5 \text{sin}(x)$. Product Rule: $(x^5)' \times \cos(x) + x^5 \times (\cos(x))'$.

22. What is $d/dx$ for $h(x) = x^3 \times \text{ln}(x)$ using the Product Rule?

    $3x^2 \text{ln}(x) + x^2$. Product Rule: $(x^3)' \times \ln(x) + x^3 \times (\ln(x))'$.

23. Determine the derivative: $y = (x^4 + x)(x^3 - 2)$.

    $7x^6 - 2x^3 + 3x^4 - 2$. Product Rule: $(4x^3 + 1) \times (x^3 - 2) + (x^4 + x) \times 3x^2$.

24. Which rule is used to differentiate the product of two functions?

    The Product Rule. Essential rule for differentiating products of functions.

25. Evaluate the derivative: $h(x) = x^2 e^x$ using the Product Rule.

    $2x e^x + x^2 e^x$. Product Rule: $(x^2)' \times e^x + x^2 \times (e^x)'$.

26. What is the derivative of $y = (x^4)(e^x)$ using Product Rule?

    $4x^3 e^x + x^4 e^x$. Product Rule: $(x^4)' \times e^x + x^4 \times (e^x)'$.

27. Determine the derivative: $f(x) = (x^3)(\text{tan}(x))$. Use Product Rule.

    $3x^2 \text{tan}(x) + x^3 \text{sec}^2(x)$. Product Rule: $(x^3)' \times \tan(x) + x^3 \times (\tan(x))'$.

28. Identify the derivative of $y = (3x + 2)(x^3 - 1)$ using the Product Rule.

    $9x^3 + 6x^2 - 3x - 2$. Product Rule: $3 \times (x^3 - 1) + (3x + 2) \times 3x^2$.

29. Evaluate the derivative: $y = (x^3 + 2)(x^2 - 1)$ using Product Rule.

    $5x^4 - 3x^2 + 4x - 2$. Product Rule: $(3x^2) \times (x^2 - 1) + (x^3 + 2) \times 2x$.

30. Is the Product Rule used for $f(x) = (x + 1)(x^2 - 1)$?

    Yes, it is applicable. Product of two functions requires the Product Rule.

31. Find the derivative: $y = (3x^2 + x)(x^2 - x)$ using Product Rule.

    $9x^3 - 4x^2 + x$. Product Rule: $(6x + 1) \times (x^2 - x) + (3x^2 + x) \times (2x - 1)$.

32. Find the derivative of $y = (x^3 + 2x)(x^2 - 3)$ using Product Rule.

    $5x^4 - 9x^2 + 4x - 6$. Product Rule: $(3x^2 + 2) \times (x^2 - 3) + (x^3 + 2x) \times 2x$.

33. Calculate the derivative of $h(x) = x \text{ln}(x)$.

    $1 \times \text{ln}(x) + \frac{x}{x}$. Product Rule: $1 \times \ln(x) + x \times \frac{1}{x}$.

34. Find the derivative of $f(x) = (x^2 + 1)(\text{sin}(x))$.

    $2x \times \text{sin}(x) + (x^2 + 1) \times \text{cos}(x)$. Product Rule: $(2x) \times \sin(x) + (x^2 + 1) \times (\sin(x))'$.

35. What is the derivative of $h(x) = x^2 \times \text{sec}(x)$ using Product Rule?

    $2x \times \text{sec}(x) + x^2 \times \text{sec}(x)\text{tan}(x)$. Product Rule: $(x^2)' \times \sec(x) + x^2 \times (\sec(x))'$.

36. Calculate the derivative for $y = (\text{sin}(x))(x^2)$ using Product Rule.

    $2x \times \text{sin}(x) + x^2 \times \text{cos}(x)$. Product Rule applied with order switched: same result.

37. Find the derivative of $y = (x^2 + 3x + 1)(x - 2)$.

    $3x^2 + 2x - 5$. Product Rule: $(2x + 3) \times (x - 2) + (x^2 + 3x + 1) \times 1$.

38. Find $d/dx$ of $y = x(x^2 + 1)$ using Product Rule.

    $3x^2 + 1$. Product Rule: $1 \times (x^2 + 1) + x \times 2x$.

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