Introducing Calculus - AP Calculus BC

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

   $\frac{f'(x)}{f(x)}$. Use the Chain Rule with the natural logarithm function.

2. State the Power Rule for derivatives.

   If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Multiply by the exponent, then reduce the exponent by 1.

3. What is the derivative of $\text{e}^{f(x)}$?

   $f'(x) \text{e}^{f(x)}$. Use the Chain Rule with the exponential function.

4. State the Chain Rule for derivatives.

   If $f(g(x))$, then $(f \circ g)'(x) = f'(g(x))g'(x)$. Derivative of outer function times derivative of inner function.

5. Which rule is used to find the derivative of a quotient of two functions?

   The Quotient Rule. Use this when differentiating quotients of two functions.

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

   $-\text{csc}(x)\text{cot}(x)$. This is the standard derivative formula for cosecant.

7. Identify the derivative of $f(x) = \frac{1}{\text{sin}(x)}$.

   $f'(x) = -\frac{\text{cos}(x)}{\text{sin}^2(x)}$. Rewrite as $\csc(x)$ and use the derivative formula.

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

   $-\text{csc}^2(x)$. This is the standard derivative formula for cotangent.

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

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

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

    $f'(x) = \frac{1}{x}$. The natural logarithm's derivative is the reciprocal function.

11. Find the derivative of $f(x) = 5x^3 - 4x + 7$.

    $f'(x) = 15x^2 - 4$. Apply the Power Rule to each term: $3 \cdot 5x^2 - 4 \cdot 1 + 0$.

12. Identify the derivative of the inverse function $f^{-1}(x)$.

    $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$. The derivative of an inverse function uses this reciprocal relationship.

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

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

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

    $\text{sec}(x)\text{tan}(x)$. This is the standard derivative formula for secant.

15. Find the derivative of $f(x) = x \text{e}^x$ using the Product Rule.

    $f'(x) = \text{e}^x + x \text{e}^x$. Apply Product Rule: $1 \cdot e^x + x \cdot e^x = e^x(1 + x)$.

16. What is the geometric interpretation of a derivative?

    The slope of the tangent line to the curve at a point. The derivative gives the instantaneous slope at any point.

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

    $nx^{n-1}$. This is the Power Rule: bring down the exponent and subtract 1.

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

    $f'(x) = 2x \text{sec}^2(x^2)$. Apply Chain Rule: derivative of $\tan(u)$ is $\sec^2(u) \cdot u'$ where $u = x^2$.

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

    $f'(x) = 4x^3 - 4x + 1$. Apply Power Rule to each term: $4x^3 - 4x + 1$.

20. Which rule is used to find the derivative of a product of two functions?

    The Product Rule. Use this when differentiating products of two functions.

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

    $\cos(x)$. The derivative of sine is cosine.

22. Identify the derivative of $f(x) = \text{sin}^2(x)$.

    $f'(x) = 2\text{sin}(x)\text{cos}(x)$. Apply Chain Rule: $2\sin(x) \cdot \cos(x)$ or equivalently $\sin(2x)$.

23. What is the derivative of $\text{arcsin}(x)$?

    $\frac{1}{\sqrt{1-x^2}}$. This is the standard derivative formula for inverse sine.

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

    -$\sin(x)$. The derivative of cosine is negative sine.

25. State the Product Rule for derivatives.

    If $u(x)$ and $v(x)$, then $(uv)' = u'v + uv'$. First function times derivative of second plus second times derivative of first.

26. Which rule is used for differentiating compositions of functions?

    The Chain Rule. Use this rule for composite functions like $f(g(x))$.

27. State the derivative of $\text{arctan}(x)$.

    $\frac{1}{1+x^2}$. This is the standard derivative formula for inverse tangent.

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

    $f'(x) = \frac{2x}{x^2 + 1}$. Apply Chain Rule: $\frac{1}{x^2 + 1} \cdot 2x$.

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

    $f'(x) = 2x \text{e}^{x^2}$. Apply Chain Rule: derivative of $e^u$ is $e^u \cdot u'$ where $u = x^2$.

30. State the Quotient Rule for derivatives.

    If $u(x)$ and $v(x)$, then $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$. Low times derivative of high minus high times derivative of low, over low squared.

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

    $f'(x) = -\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply the Power Rule.

32. State the formula for the derivative of $f(x)$ using limits.

    $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. This is the formal limit definition of the derivative.

33. Identify the derivative of a constant function.

    Zero. Constants have no rate of change, so their derivative is zero.

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

    $f'(x) = 2 \text{cos}(2x)$. Apply Chain Rule: derivative of $\sin(u)$ is $\cos(u) \cdot u'$ where $u = 2x$.

35. Find the derivative of $f(x) = 3x^2 + 2x + 1$.

    $f'(x) = 6x + 2$. Apply the Power Rule to each term separately.

36. Find the derivative of $f(x) = x e^x$ using the Product Rule.

    $f'(x) = e^x + x e^x$. Apply Product Rule: $1 \cdot e^x + x \cdot e^x = e^x(1 + x)$.

37. Identify the derivative of $f(x) = \text{cos}(3x)$.

    $f'(x) = -3 \text{sin}(3x)$. Apply Chain Rule: derivative of $\cos(u)$ is $-\sin(u) \cdot u'$ where $u = 3x$.

38. Which function's derivative is $e^x$?

    $e^x$. The exponential function $e^x$ is its own derivative.

39. What is the derivative of $\text{arccos}(x)$?

    $-\frac{1}{\sqrt{1-x^2}}$. This is the standard derivative formula for inverse cosine.

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

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

41. What is the definition of a derivative?

    The limit of the average rate of change as the interval approaches zero. This captures the instantaneous rate of change concept.

Study these flashcards