Introducing Calculus - AP Calculus AB

1. What is the derivative of $f(x) = a^x$, where $a > 0$?

   $f'(x) = a^x \text{ln}(a)$. For exponential with base $a$, multiply by $\ln(a)$.

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

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

3. Differentiate $f(x) = \text{arccsc}(x)$.

   $f'(x) = -\frac{1}{|x|\text{sqrt}(x^2-1)}$. Similar to arcsec but with negative sign.

4. State the limit definition of a derivative.

   $f'(x) = \frac{d}{dx}f(x) = \text{lim}_{h \to 0} \frac{f(x+h) - f(x)}{h}$. This is the formal limit definition using the difference quotient.

5. Identify the derivative of $f(x) = x^2$.

   $f'(x) = 2x$. Using the power rule: bring down the exponent and subtract 1.

6. Apply the Product Rule to $f(x) = x^2 \text{sin}(x)$.

   $f'(x) = 2x\text{sin}(x) + x^2\text{cos}(x)$. Apply product rule: $u = x^2$, $v = \sin(x)$.

7. Differentiate $f(x) = \text{arctan}(x)$.

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

8. What is the Chain Rule for differentiation?

   If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x))g'(x)$. This rule handles composite functions: outer derivative times inner.

9. Identify the derivative of $f(x) = \text{arccot}(x)$.

   $f'(x) = -\frac{1}{1+x^2}$. Similar to arctan but with negative sign.

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

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

11. Apply the Power Rule to $f(x) = x^5$.

    $f'(x) = 5x^4$. Using power rule: $5x^{5-1} = 5x^4$.

12. Differentiate $f(x) = \text{tan}(x)$.

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

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

    $f'(x) = -\text{csc}^2(x)$. The derivative of cotangent is negative cosecant squared.

14. Differentiate $f(x) = 7x + 4$.

    $f'(x) = 7$. Linear term becomes its coefficient; constant disappears.

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

    $f'(x) = \frac{1}{x}$. This is a fundamental derivative of logarithmic functions.

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

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

17. What is the definition of a derivative?

    The derivative is the instantaneous rate of change of a function. This is the fundamental concept of calculus.

18. What is the derivative of $f(x) = \text{arcsec}(x)$?

    $f'(x) = \frac{1}{|x|\text{sqrt}(x^2-1)}$. The derivative includes absolute value due to domain restrictions.

19. Identify the derivative of $f(x) = \text{e}^{3x}$.

    $f'(x) = 3\text{e}^{3x}$. Chain rule: $e^{3x}$ times derivative of $3x$.

20. What does the derivative tell us about a function at a point?

    The slope of the tangent line to the function at that point. The derivative measures the steepness of the curve at any point.

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

    $f'(x) = 3x^2 - 5$. Apply power rule term by term: $3x^2 - 5 + 0$.

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

    $f'(x) = -\text{csc}(x)\text{cot}(x)$. This follows from the chain rule applied to $\csc(x) = \frac{1}{\sin(x)}$.

23. What is the derivative of a constant function $f(x) = c$?

    $f'(x) = 0$. Constants have no change, so their rate of change is zero.

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

    $f'(x) = -\frac{1}{\text{sqrt}(1-x^2)}$. Similar to arcsin but with negative sign.

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

    $f'(x) = \text{sec}(x)\text{tan}(x)$. This follows from the chain rule applied to $\sec(x) = \frac{1}{\cos(x)}$.

26. What is the derivative of $f(x) = e^x$?

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

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

    $f'(x) = \frac{1}{x \text{ln}(a)}$. For logarithm base $a$, include the factor $\frac{1}{\ln(a)}$.

28. Apply the Chain Rule to $f(x) = (3x^2 + 2)^5$.

    $f'(x) = 5(3x^2 + 2)^4 \times 6x$. Outer function derivative is $5(3x^2 + 2)^4$, inner is $6x$.

29. What is the Quotient Rule for differentiation?

    If $\frac{u}{v}$, then $\frac{vu' - uv'}{v^2}$.. For quotients: bottom times top's derivative minus top times bottom's derivative, all over bottom squared.

30. Differentiate $f(x) = \text{arcsin}(x)$.

    $f'(x) = \frac{1}{\text{sqrt}(1-x^2)}$. This is the standard derivative formula for inverse sine.

31. Differentiate $f(x) = \text{ln}(3x)$ using the Chain Rule.

    $f'(x) = \frac{3}{3x} = \frac{1}{x}$. Chain rule: derivative of $\ln(u)$ is $\frac{u'}{u}$.

32. What is the Power Rule for differentiation?

    If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.. This is the fundamental differentiation rule for polynomial terms.

33. Find the derivative of $f(x) = 3x^3$.

    $f'(x) = 9x^2$. Apply power rule to $x^3$: $3 \cdot 3x^2 = 9x^2$.

34. What is the Product Rule for differentiation?

    If $u(x)v(x)$, then $uv' + vu'$.. For products: first times derivative of second plus second times derivative of first.

35. Differentiate $f(x) = \text{e}^{-x^2}$ using the Chain Rule.

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

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