The Product Rule - AP Calculus BC

1. Determine the derivative of $f(x) = x^2 \times \text{e}^x$ using the Product Rule.

   $2x \times e^x + x^2 \times e^x$. Factor out $x e^x$ to get $x e^x(2 + x)$.

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

   $2x \times \text{ln}(x) + x$. Apply $(fg)' = f'g + fg'$ with $f = x^2$ and $g = \ln(x)$.

3. Identify the derivative of $f(x) = e^x \times \text{sin}(x)$ using the Product Rule.

   $e^x \times \text{sin}(x) + e^x \times \text{cos}(x)$. Use $(fg)' = f'g + fg'$ with derivatives of $e^x$ and $\sin(x)$.

4. What is the derivative of $f(x) = x^4 \times \text{tan}(x)$ using the Product Rule?

   $4x^3 \times \text{tan}(x) + x^4 \times \text{sec}^2(x)$. Factor out $x^3$ to get $x^3(4\tan(x) + x\sec^2(x))$.

5. What is the derivative of $f(x) = x^2 \times \text{e}^{3x}$ using the Product Rule?

   $2x \times \text{e}^{3x} + 3x^2 \times \text{e}^{3x}$. Factor out $x e^{3x}$ to get $x e^{3x}(2 + 3x)$.

6. Apply the Product Rule to find $f'(x)$ for $f(x) = x \times \text{e}^x$.

   $e^x + x \times e^x$. Factor out $e^x$ to get $e^x(1 + x)$.

7. Which option correctly applies the Product Rule to $f(x) = x^3 \times \text{cos}(x)$?

   $3x^2 \times \text{cos}(x) - x^3 \times \text{sin}(x)$. Apply the product rule: derivative of first times second plus first times derivative of second.

8. Identify the derivative of $f(x) = x^5 \times \text{e}^{2x}$ using the Product Rule.

   $5x^4 \times e^{2x} + 2x^5 \times e^{2x}$. Factor out $x^4 e^{2x}$ to get $x^4 e^{2x}(5 + 2x)$.

9. Compute the derivative of $f(x) = x \times \text{cosh}(x)$ using the Product Rule.

   $\text{cosh}(x) + x \times \text{sinh}(x)$. Apply $(fg)' = f'g + fg'$ with hyperbolic function derivatives.

10. Derive the function $f(x) = x^2 \times \text{e}^{2x}$ using the Product Rule.

    $2x \times e^{2x} + 2x^2 \times e^{2x}$. Factor out $2x e^{2x}$ to get $2x e^{2x}(1 + x)$.

11. Using the Product Rule, what is the derivative of $f(x) = x \times \text{log}(x)$?

    $\text{log}(x) + 1$. Use the fact that $\frac{d}{dx}[x \log(x)] = \log(x) + 1$.

12. Compute the derivative using the Product Rule for $y = x \times \text{cos}(x)$.

    $\text{cos}(x) - x \times \text{sin}(x)$. Apply the product rule with derivatives $1$ and $-\sin(x)$.

13. Calculate the derivative using the Product Rule for $f(x) = 2x \times \text{sin}(x)$.

    $2 \times \text{sin}(x) + 2x \times \text{cos}(x)$. Factor out $2$ to get $2(\sin(x) + x\cos(x))$.

14. Derive the function $f(x) = 5x \times \text{e}^{-x}$ using the Product Rule.

    $5e^{-x} - 5x \times e^{-x}$. Factor out $5e^{-x}$ to get $5e^{-x}(1 - x)$.

15. Find the derivative of $f(x) = (x^2 + 1)(x^3 - x)$ using the Product Rule.

    $(2x)(x^3 - x) + (x^2 + 1)(3x^2 - 1)$. Apply product rule to each factor and combine the terms.

16. Calculate the derivative using the Product Rule for $y = (1 - x) \times (2 + x)$.

    $-(2 + x) + (1 - x)$. Expand and simplify to get $-1 - 2x$.

17. What is the derivative of $f(x) = x \times \text{sin}(x)$ using the Product Rule?

    $\text{sin}(x) + x \times \text{cos}(x)$. Apply the product rule with derivatives $1$ and $\cos(x)$.

18. What is the derivative of $f(x) = x^3 \times \text{sin}(x)$ using the Product Rule?

    $3x^2 \times \text{sin}(x) + x^3 \times \text{cos}(x)$. Factor out $x^2$ to get $x^2(3\sin(x) + x\cos(x))$.

19. Find the derivative using the Product Rule for $f(x) = x \times \text{cosh}(x)$.

    $\text{cosh}(x) + x \times \text{sinh}(x)$. Apply the product rule with hyperbolic function derivatives.

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

    $2x \times \text{cos}(x) - x^2 \times \text{sin}(x)$. Factor out $x$ to get $x(2\cos(x) - x\sin(x))$.

21. What is the derivative of $f(x) = x^2 \times \text{e}^{-x}$ using the Product Rule?

    $2x \times e^{-x} - x^2 \times e^{-x}$. Factor out $x e^{-x}$ to get $x e^{-x}(2 - x)$.

22. Apply the Product Rule to $f(x) = \text{sin}(x) \times \text{cos}(x)$.

    $\text{cos}^2(x) - \text{sin}^2(x)$. This simplifies to $\cos(2x)$ using the double angle identity.

23. What is the result of applying the Product Rule to $y = x \times e^{2x}$?

    $e^{2x} + 2xe^{2x}$. Factor out $e^{2x}$ to get $e^{2x}(1 + 2x)$ after applying the rule.

24. What is the derivative of $f(x) = x^3 \times \ln(x)$ using the Product Rule?

    $3x^2 \times \ln(x) + x^2$. Use $(fg)' = f'g + fg'$ where $\frac{d}{dx}[x^3] = 3x^2$ and $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$.

25. Identify the derivative of $f(x) = x^3 \times \text{e}^x$ using the Product Rule.

    $3x^2 \times e^x + x^3 \times e^x$. Factor out $x^2 e^x$ to get $x^2 e^x(3 + x)$.

26. What is the derivative of $f(x) = x \times \text{tan}(x)$ using the Product Rule?

    $\text{tan}(x) + x \times \text{sec}^2(x)$. Apply the product rule with derivatives $1$ and $\sec^2(x)$.

27. Calculate the derivative of $f(x) = (x + 1) \times (x - 2)$ using the Product Rule.

    $(x - 2) + (x + 1)$. Simplifies to $2x - 1$ when expanded and combined.

28. Using the Product Rule, find the derivative for $y = x^2 \times \text{tan}(x)$.

    $2x \times \text{tan}(x) + x^2 \times \text{sec}^2(x)$. Factor out $x$ to get $x(2\tan(x) + x\sec^2(x))$.

29. State the formula for the Product Rule in calculus.

    $(fg)' = f'g + fg'$. The fundamental formula where each function's derivative multiplies the other function.

30. What is the derivative of $f(x) = x^3 \times \text{ln}(x)$ using the Product Rule?

    $3x^2 \times \text{ln}(x) + x^2$. Use $(fg)' = f'g + fg'$ where $\frac{d}{dx}[x^3] = 3x^2$ and $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$.

31. Calculate the derivative using the Product Rule for $y = (2x) \times \text{ln}(x)$.

    $2 \times \text{ln}(x) + \frac{2x}{x}$. Simplifies to $2\ln(x) + 2$ since $\frac{2x}{x} = 2$.

32. Which is the correct application of the Product Rule to $f(x) = 3x \times \text{ln}(x)$?

    $3 \times \text{ln}(x) + \frac{3x}{x}$. Simplifies to $3 \ln(x) + 3$ since $\frac{3x}{x} = 3$.

33. Apply the Product Rule to find $f'(x)$ for $f(x) = (x + 1) \times \text{e}^x$.

    $e^x + (x + 1) \times e^x$. Factor out $e^x$ to get $e^x(1 + x + 1) = e^x(2 + x)$.

34. Determine the derivative of $y = (3x) \times \text{e}^x$ using the Product Rule.

    $3e^x + 3x \times e^x$. Factor out $3e^x$ to simplify to $3e^x(1 + x)$.

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