The Quotient Rule - AP Calculus BC

1. What is the derivative of $\frac{x^4}{x^2}$ using the Quotient Rule?

   $\frac{x^2 \cdot 4x^3 - x^4 \cdot 2x}{(x^2)^2}$. Could simplify to $x^2$ first, then derivative is $2x$.

2. Identify $u$ and $v$ for $\frac{x+1}{x^2}$ in the Quotient Rule.

   $u = x+1$, $v = x^2$. Linear numerator, quadratic denominator.

3. What is the derivative of $\frac{\ln x}{x}$ using the Quotient Rule?

   $\frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x^2}$. Quotient rule with logarithmic numerator.

4. Determine $u$ and $v$ for $\frac{2x}{x^3}$ using the Quotient Rule.

   $u = 2x$, $v = x^3$. Linear numerator, cubic denominator.

5. Find $u'$ for $u = x^3$ in the Quotient Rule.

   $u' = 3x^2$. Cubic function derivative.

6. Find $u'$ when $u = x^4$ in the Quotient Rule.

   $u' = 4x^3$. Power rule applied to fourth power.

7. What is the derivative of $\frac{x^2}{x+1}$ using the Quotient Rule?

   $\frac{(x+1)\cdot 2x - x^2 \cdot 1}{(x+1)^2}$. Applying quotient rule with $u = x^2$, $v = x+1$.

8. Find the derivative of $\frac{x^3}{3x+2}$ using the Quotient Rule.

   $\frac{(3x+2)\cdot 3x^2 - x^3 \cdot 3}{(3x+2)^2}$. Quotient rule with cubic numerator.

9. What is $v'$ if $v = x^3$ in the Quotient Rule?

   $v' = 3x^2$. Power rule applied to cubic function.

10. Find the derivative of $\frac{3x}{x^2+1}$ using the Quotient Rule.

    $\frac{(x^2+1)\cdot 3 - 3x \cdot 2x}{(x^2+1)^2}$. Quotient rule with $u = 3x$, $v = x^2+1$.

11. What is $u'$ if $u = e^x$ in the Quotient Rule?

    $u' = e^x$. Exponential function derivative.

12. What is the derivative of $\frac{x^2}{x^2+1}$ using the Quotient Rule?

    $\frac{(x^2+1)\cdot 2x - x^2 \cdot 2x}{(x^2+1)^2}$. Quotient rule applied to rational function.

13. Find $v'$ for $v = x^3$ in the Quotient Rule.

    $v' = 3x^2$. Cubic function derivative.

14. What is $v'$ if $v = e^x$ in the Quotient Rule?

    $v' = e^x$. Exponential function is its own derivative.

15. Find $u'$ for $u = \sin x$ in the Quotient Rule.

    $u' = \cos x$. Derivative of sine function.

16. What is the derivative of $\frac{\cos x}{x}$ using the Quotient Rule?

    $\frac{x \cdot (-\sin x) - \cos x \cdot 1}{x^2}$. Quotient rule with $u = \cos x$, $v = x$.

17. Determine $u$ and $v$ for $\frac{1}{e^x}$ using the Quotient Rule.

    $u = 1$, $v = e^x$. Constant numerator, exponential denominator.

18. Identify $u$ and $v$ for $\frac{e^x}{\sin x}$ in the Quotient Rule.

    $u = e^x$, $v = \sin x$. Numerator and denominator identification for quotient rule.

19. Identify $u$ and $v$ for $\frac{x^5}{\ln x}$ in the Quotient Rule.

    $u = x^5$, $v = \ln x$. Power and logarithm function identification.

20. Find the derivative of $\frac{1}{x}$ using the Quotient Rule.

    $\frac{x \cdot 0 - 1 \cdot 1}{x^2}$. Constant numerator means $u' = 0$.

21. What is the derivative of $\frac{\ln x}{e^x}$ using the Quotient Rule?

    $\frac{e^x \cdot \frac{1}{x} - \ln x \cdot e^x}{(e^x)^2}$. Quotient rule with mixed functions.

22. What is $v'$ if $v = \sin x$ in the Quotient Rule?

    $v' = \cos x$. Derivative of sine function.

23. Find $u'$ for $u = \cos x$ in the Quotient Rule.

    $u' = -\sin x$. Derivative of cosine is negative sine.

24. Find $u'$ for $u = \tan x$ in the Quotient Rule.

    $u' = \sec^2 x$. Derivative of tangent function.

25. State the formula for the Quotient Rule in calculus.

    $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2}$. Standard derivative formula for quotient of two functions.

26. Identify $u$ and $v$ for $\frac{x^2+1}{x}$ in the Quotient Rule.

    $u = x^2+1$, $v = x$. Quadratic numerator, linear denominator.

27. Find $v'$ for $v = x^2+1$ in the Quotient Rule.

    $v' = 2x$. Derivative of sum with constant term.

28. What is the derivative of $\frac{\sin x}{x^2}$ using the Quotient Rule?

    $\frac{x^2 \cdot \cos x - \sin x \cdot 2x}{x^4}$. Quotient rule with trigonometric numerator.

29. What is $v'$ if $v = \tan x$ in the Quotient Rule?

    $v' = \sec^2 x$. Derivative of tangent function.

30. Determine $u$ and $v$ for $\frac{1}{x^2}$ using the Quotient Rule.

    $u = 1$, $v = x^2$. Constant numerator, quadratic denominator.

31. What is the derivative of $\frac{x}{e^x}$ using the Quotient Rule?

    $\frac{e^x \cdot 1 - x \cdot e^x}{(e^x)^2}$. Quotient rule with exponential denominator.

32. What is $u'$ if $u = \ln x$ in the Quotient Rule?

    $u' = \frac{1}{x}$. Natural logarithm derivative.

33. Choose $u$ and $v$ for $\frac{\tan x}{\cos x}$ in the Quotient Rule.

    $u = \tan x$, $v = \cos x$. Numerator and denominator for trigonometric quotient.

34. Identify $u$ and $v$ for $\frac{e^x}{x}$ in the Quotient Rule.

    $u = e^x$, $v = x$. Exponential numerator, linear denominator.

35. Find the derivative of $\frac{2x+3}{x^2}$ using the Quotient Rule.

    $\frac{x^2 \cdot 2 - (2x+3) \cdot 2x}{x^4}$. Quotient rule setup for rational function.

36. What is the derivative of $\frac{x}{x^2}$ using the Quotient Rule?

    $\frac{x^2 \cdot 1 - x \cdot 2x}{x^4}$. Quotient rule simplifies to $\frac{1}{x}$ then derivative.

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