Derivative Notation - AP Calculus AB

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

   $f'(x) = 2\text{sin}(x)\text{cos}(x)$. Chain rule with power of sine function.

2. What does $D_x[f(x)]$ represent?

   The derivative of $f(x)$ with respect to $x$. Operator notation where $D_x$ indicates differentiation with respect to $x$.

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

   $f'(x) = \frac{2}{x}$. Use property $\ln(x^2) = 2\ln(x)$ then differentiate.

4. What is the derivative of $\text{csc}(x)$?

   $-\text{csc}(x)\text{cot}(x)$. Derivative of cosecant is negative cosecant cotangent.

5. What is the derivative of a sum $f(x) + g(x)$?

   $f'(x) + g'(x)$. Derivative of sum equals sum of derivatives.

6. What does $f'(x)$ represent graphically?

   The slope of the tangent line to $f(x)$ at $x$. Derivative gives instantaneous rate of change at each point.

7. State the Power Rule for derivatives.

   $\frac{d}{dx}x^n = nx^{n-1}$. Fundamental derivative rule for power functions.

8. Differentiate $f(x) = 5\text{sin}(x)$.

   $f'(x) = 5\text{cos}(x)$. Constant factor rule: $\frac{d}{dx}[cf(x)] = cf'(x)$.

9. What is the derivative of $f(x) = 3^x$?

   $f'(x) = 3^x\text{ln}(3)$. Exponential with base $a$ involves $\ln(a)$ factor.

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

    $e^x$. Exponential function $e^x$ is its own derivative.

11. Identify the derivative notation for $y$ with respect to $x$.

    $\frac{dy}{dx}$. Leibniz notation for derivative of $y$ with respect to $x$.

12. State the derivative of $f(x)$ in prime notation.

    $f'(x)$. Prime notation for first derivative of function $f$.

13. What is the derivative of a quotient $\frac{f(x)}{g(x)}$?

    $\frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$. Quotient rule for differentiating ratios of functions.

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

    $\cos(x)$. Derivative of sine is cosine.

15. What is the derivative of a product $f(x)g(x)$?

    $f'(x)g(x) + f(x)g'(x)$. Product rule for differentiating products of functions.

16. What is the derivative of a difference $f(x) - g(x)$?

    $f'(x) - g'(x)$. Derivative of difference equals difference of derivatives.

17. State the derivative of $f(x) = x\text{ln}(x)$.

    $f'(x) = 1 + \text{ln}(x)$. Product rule applied to $x$ and $\ln(x)$.

18. What is the derivative of $\text{log}_a(x)$?

    $\frac{1}{x\text{ln}(a)}$. Logarithm base $a$ derivative involves natural log of base.

19. What is the derivative of $\text{cos}(x)$?

    $-\text{sin}(x)$. Derivative of cosine is negative sine.

20. Differentiate $f(x) = x^5 - 4x^3 + 2$.

    $f'(x) = 5x^4 - 12x^2$. Apply power rule to each polynomial term.

21. Find $f'(x)$ for $f(x) = \text{cos}(2x)$.

    $f'(x) = -2\text{sin}(2x)$. Chain rule with cosine and linear inner function.

22. What is the derivative of $\text{cot}(x)$?

    $-\text{csc}^2(x)$. Derivative of cotangent is negative cosecant squared.

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

    $f'(x) = 2\text{e}^{2x}$. Chain rule: derivative of outer times derivative of inner.

24. Find the derivative of $g(x) = \frac{2}{x}$.

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

25. Determine the derivative of $h(x) = x^3 + 4x$.

    $h'(x) = 3x^2 + 4$. Power rule applied to each term.

26. What is the definition of the derivative of a function at a point $x=a$?

    $f'(a) = \frac{d}{dx}f(x) \bigg|_{x=a}$. Notation shows derivative of $f$ evaluated at point $a$.

27. What is the derivative of $\log_a(x)$?

    $\frac{1}{x \ln(a)}$. Logarithm base $a$ derivative involves natural log of base.

28. What is the derivative of $\text{tan}(x)$?

    $\text{sec}^2(x)$. Derivative of tangent is secant squared.

29. What is the Chain Rule in derivative notation?

    $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Formula for differentiating composite functions.

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

    $f'(x) = 6x + 5$. Apply power rule to each term separately.

31. What is the derivative of $\text{ln}(x)$?

    $\frac{1}{x}$. Natural logarithm derivative is reciprocal function.

32. Differentiate $f(x) = \frac{x^2}{x+1}$.

    $f'(x) = \frac{x(x+2)}{(x+1)^2}$. Apply quotient rule to rational function.

33. How is the derivative of $y$ with respect to $x$ written using Leibniz's notation?

    $\frac{dy}{dx}$. Standard Leibniz differential notation for derivatives.

34. What is the derivative of $\text{sec}(x)$?

    $\text{sec}(x)\text{tan}(x)$. Derivative of secant involves secant times tangent.

35. Identify the derivative of $f(x) = \text{e}^{-x}$.

    $f'(x) = -\text{e}^{-x}$. Chain rule with exponential function and negative exponent.

36. What is the derivative of $x^n$ where $n$ is a constant?

    $nx^{n-1}$. Power rule: bring down exponent, reduce power by one.

37. Differentiate $f(x) = 5\sin(x)$.

    $f'(x) = 5\cos(x)$. Constant factor rule: $\frac{d}{dx}[cf(x)] = cf'(x)$.

38. What is the derivative of $\text{sin}(x)$?

    $\text{cos}(x)$. Derivative of sine is cosine.

39. What is the limit definition of the derivative of $f(x)$?

    $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. Standard limit definition using difference quotient as $h$ approaches zero.

40. Identify the derivative of $\frac{1}{x}$.

    $-\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply power rule.

41. What is the derivative of a constant function $c$?

    $0$. Constants have zero rate of change.

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