Change in Tandem - AP Precalculus

1. What is the formula for $\frac{d}{dx}(\tan(x))$?

   $\sec^2(x)$. Standard derivative formula for tangent function.

2. What is the derivative of $\frac{u}{v}$ using the Quotient Rule?

   $\frac{u'v - uv'}{v^2}$. Quotient rule: top derivative times bottom minus top times bottom derivative, over bottom squared.

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

   $-\sin(x)$. Standard derivative formula for cosine function.

4. Find the derivative of $e^{3x}$.

   $3e^{3x}$. Chain rule: derivative of $e^u$ is $e^u \cdot u'$, where $u = 3x$.

5. Identify the derivative of $\ln(x)$.

   $\frac{1}{x}$. Standard derivative formula for natural logarithm.

6. Determine $\frac{d}{dx}(\ln(x^2))$.

   $\frac{2}{x}$. Using chain rule with $\ln(u)$ where $u = x^2$, so $\frac{u'}{u} = \frac{2x}{x^2}$.

7. What is the derivative of $\sin^2(x)$ using the Chain Rule?

   $2\sin(x)\cos(x)$. Chain rule with power rule: $2\sin(x) \cdot \cos(x)$.

8. What is the derivative of $\ln(x^3)$?

   $\frac{3}{x}$. Using logarithm property: $\ln(x^3) = 3\ln(x)$, so derivative is $\frac{3}{x}$.

9. Determine $\frac{d}{dx}(\tan(x))$.

   $\sec^2(x)$. Standard derivative formula for tangent function.

10. Identify the derivative of $\sin(g(x))$ using the Chain Rule.

    $\cos(g(x)) \cdot g'(x)$. Derivative of sine is cosine, multiplied by derivative of inner function.

11. Determine $\frac{d}{dx}(\cot(x))$.

    $-\csc^2(x)$. Standard derivative formula for cotangent function.

12. Calculate $\frac{d}{dx}(5x^4)$.

    $20x^3$. Constant factor rule: multiply derivative by the constant 5.

13. What is the derivative of a constant $c$?

    0. Constants have zero rate of change.

14. Determine $\frac{d}{dx}(\sqrt{x})$.

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

15. Identify the derivative of $\cos^{-1}(x)$.

    $-\frac{1}{\sqrt{1-x^2}}$. Standard derivative formula for inverse cosine function.

16. State the Power Rule for differentiation.

    $\frac{d}{dx}(x^n) = nx^{n-1}$. Bring down the exponent and reduce the power by one.

17. Calculate $\frac{d}{dx}(x^5 + x^3)$.

    $5x^4 + 3x^2$. Apply power rule to each term separately.

18. What is the derivative of $x^3$?

    $3x^2$. Power rule: bring down exponent 3, reduce power to 2.

19. What is the derivative of $u \cdot v$ using the Product Rule?

    $u'v + uv'$. Apply product rule to find derivative of product of functions.

20. Calculate $\frac{d}{dx}(\frac{1}{x+2})$.

    $-\frac{1}{(x+2)^2}$. Chain rule: derivative of $u^{-1}$ is $-u^{-2} \cdot u'$.

21. Find the derivative of $\csc(x)$.

    $-\csc(x)\cot(x)$. Standard derivative formula for cosecant function.

22. What is the formula for the derivative of $f(g(x))$ using the Chain Rule?

    $f'(g(x)) \cdot g'(x)$. Apply outer function derivative at inner function, then multiply by inner derivative.

23. Find the derivative of $\ln(e^x)$.

    $1$. Since $\ln(e^x) = x$, the derivative is simply 1.

24. Find the derivative of $x^{\frac{1}{2}}$.

    $\frac{1}{2}x^{-\frac{1}{2}}$. Power rule with fractional exponent: $\frac{1}{2} \cdot x^{-1/2}$.

25. State the Chain Rule for differentiation.

    $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Multiplies derivatives when functions are composed inside each other.

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

    $\sec(x)\tan(x)$. Standard derivative formula for secant function.

27. Calculate $\frac{d}{dx}(\cos(5x))$.

    $-5\sin(5x)$. Derivative of cosine is negative sine, times derivative of inner function.

28. Find the derivative of $\frac{1}{x}$.

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

29. Determine $\frac{d}{dx}(\sin^{-1}(x))$.

    $\frac{1}{\sqrt{1-x^2}}$. Standard derivative formula for inverse sine function.

30. What is the derivative of $\tan^{-1}(x)$?

    $\frac{1}{1+x^2}$. Standard derivative formula for inverse tangent function.

31. Identify the derivative of $x^{-1}$.

    $-x^{-2}$. Power rule with negative exponent: $-1 \cdot x^{-2}$.

32. What does $\frac{dy}{dx}$ represent in relation to change in tandem?

    Rate of change of $y$ with respect to $x$. Shows how fast $y$ changes as $x$ changes at any point.

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

    $\cos(x)$. Standard derivative formula for sine function.

34. State the Product Rule for differentiation.

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

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

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

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