Selecting Procedures for Calculating Derivatives - AP Calculus AB

1. Identify the derivative of $f(x) = \csc(x)$.

   $f'(x) = -\csc(x)\cot(x)$. Standard trigonometric derivative formula.

2. Differentiate $f(x) = (2x+1)^3$ using the chain rule.

   $f'(x) = 3(2x+1)^2 \cdot 2$. Apply chain rule: outer derivative times inner derivative.

3. Differentiate $f(x) = \frac{1}{x}$ using the power rule.

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

4. What is the derivative of $\ln(5x)$?

   $f'(x) = \frac{1}{x}$. Constant multiple in logarithm doesn't affect derivative.

5. Find the derivative of $f(x) = 4x^4$ using the power rule.

   $f'(x) = 16x^3$. Power rule: $4 \times 4 = 16$, exponent becomes $3$.

6. Find the derivative of $f(x) = 5x^3 + 2x - 7$.

   $f'(x) = 15x^2 + 2$. Apply power rule to each term separately.

7. What is the derivative of $\cos(x)$?

   $f'(x) = -\sin(x)$. Cosine differentiates to negative sine.

8. Identify the rule used for $f(x) = \frac{x^2 + 1}{x - 1}$.

   Quotient rule. Fraction form indicates quotient rule is needed.

9. Find the derivative of $f(x) = \frac{1}{x^3}$.

   $f'(x) = -\frac{3}{x^4}$. Rewrite as $x^{-3}$ then apply power rule.

10. What is the derivative of $e^x$?

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

11. What is the derivative of $f(x) = x^n$ using the power rule?

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

12. Find the derivative of $f(x) = \cos(3x)$.

    $f'(x) = -3\sin(3x)$. Chain rule: derivative of cosine times derivative of $3x$.

13. Differentiate $f(x) = x^{3/2}$ using the power rule.

    $f'(x) = \frac{3}{2}x^{1/2}$. Power rule: $\frac{3}{2}$ coefficient, exponent becomes $\frac{1}{2}$.

14. Differentiate $f(x) = \sqrt{x}$.

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

15. Differentiate $f(x) = \cot(x)$ using trigonometric rules.

    $f'(x) = -\csc^2(x)$. Standard trigonometric derivative formula.

16. Differentiate $f(x) = \ln(4x)$.

    $f'(x) = \frac{1}{x}$. Constant multiple in logarithm doesn't affect derivative.

17. What is the derivative of $\tan(x)$?

    $f'(x) = \sec^2(x)$. Tangent differentiates to secant squared.

18. What is the derivative of $\csc(x)$?

    $f'(x) = -\csc(x)\cot(x)$. Cosecant differentiates to negative cosecant cotangent.

19. State the quotient rule for derivatives.

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

20. State the product rule for derivatives.

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

21. Which rule is used to differentiate $f(x) = (3x^2)(\ln(x))$?

    Product rule. Two functions multiplied together requires product rule.

22. What is the derivative of $\sec(x)$?

    $f'(x) = \sec(x)\tan(x)$. Secant differentiates to secant tangent.

23. What is $f'(x)$ for $f(x) = 7$ (constant function)?

    $f'(x) = 0$. Derivative of any constant is zero.

24. Find the derivative of $f(x) = \frac{x^3}{3}$.

    $f'(x) = x^2$. Derivative of $\frac{x^3}{3}$ using power rule on numerator.

25. Differentiate $f(x) = \ln(x^2 + 1)$.

    $f'(x) = \frac{2x}{x^2 + 1}$. Chain rule with natural log and composite function.

26. Which differentiation rule applies to $f(x) = x^2 \sin(x)$?

    Product rule. Polynomial times trigonometric function needs product rule.

27. Differentiate $f(x) = \sin(2x)$.

    $f'(x) = 2\cos(2x)$. Chain rule: derivative of sine times derivative of $2x$.

28. What is the derivative of $f(x) = 3x^{-2}$?

    $f'(x) = -6x^{-3}$. Power rule: $3 \times (-2) = -6$, exponent becomes $-3$.

29. What is the derivative of $f(x) = \frac{7}{x^2}$?

    $f'(x) = -\frac{14}{x^3}$. Rewrite as $7x^{-2}$ then apply power rule.

30. What is the derivative of $\ln(x)$?

    $f'(x) = \frac{1}{x}$. Natural logarithm derivative is reciprocal function.

31. What is the derivative of $f(x) = x^{-3}$?

    $f'(x) = -3x^{-4}$. Power rule with negative exponent: $-3 \times (-1) = 3$, new power $-4$.

32. What is the derivative of $\sin(x)$?

    $f'(x) = \cos(x)$. Sine differentiates to cosine.

33. Differentiate $f(x) = \frac{7}{x}$.

    $f'(x) = -\frac{7}{x^2}$. Rewrite as $7x^{-1}$ then apply power rule.

34. Differentiate $f(x) = \sec(x)$ using trigonometric differentiation.

    $f'(x) = \sec(x)\tan(x)$. Standard trigonometric derivative formula.

35. Select the rule for $f(x) = e^{3x}$.

    Chain rule. Composite function requires chain rule for differentiation.

36. Differentiate $f(x) = 5e^{2x}$.

    $f'(x) = 10e^{2x}$. Chain rule: $5 \times e^{2x} \times 2$.

37. What is the derivative of $\cot(x)$?

    $f'(x) = -\csc^2(x)$. Cotangent differentiates to negative cosecant squared.

38. State the chain rule for derivatives.

    $(f(g(x)))' = f'(g(x))g'(x)$. Multiply derivative of outer by derivative of inner function.

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