Chain Rule - AP Calculus AB

1. Differentiate $y = (\ln(x))^3$.

   $\frac{3(\ln(x))^2}{x}$. Power rule: $3(\ln(x))^2 \times \frac{1}{x}$.

2. Differentiate $y = e^{\tan(x)}$

   $\sec^2(x) e^{\tan(x)}$. Exponential times derivative of tangent function.

3. Differentiate $y = \sin^3(x)$.

   $3\sin^2(x)\cos(x)$. Power rule: $3\sin^2(x) \times \cos(x)$.

4. Differentiate $y = \text{e}^{2x^2 + 3x}$.

   $(4x + 3)\text{e}^{2x^2 + 3x}$. Exponential function times derivative of exponent.

5. Differentiate $y = \text{sin}(\text{e}^{3x})$.

   $3\text{e}^{3x}\text{cos}(\text{e}^{3x})$. Sine composition: $\cos(\text{e}^{3x}) \times 3\text{e}^{3x}$.

6. Find $\frac{d}{dx}(\text{e}^{3x+1})$.

   $3\text{e}^{3x+1}$. Exponential function times derivative of exponent.

7. Differentiate $y = \text{cos}(5x)$.

   $-5\text{sin}(5x)$. Derivative of cosine is $-\sin$ times inner derivative.

8. Differentiate $y = \text{cos}(\text{sin}(x))$.

   $-\text{sin}(\text{sin}(x))\text{cos}(x)$. Cosine composition: $-\sin(\sin(x)) \times \cos(x)$.

9. Differentiate $y = \text{e}^{x^2 + 3}$.

   $2x\text{e}^{x^2 + 3}$. Exponential function times derivative of exponent.

10. Differentiate $y = \text{ln}(\text{e}^{2x})$.

    2. Simplifies to $\ln(\text{e}^{2x}) = 2x$.

11. Differentiate $y = \text{tan}^2(x)$.

    $2\text{tan}(x)\text{sec}^2(x)$. Power rule: $2\tan(x) \times \sec^2(x)$.

12. Differentiate $y = \frac{1}{(2x^2 + 3)^4}$.

    $-\frac{16x}{(2x^2 + 3)^5}$. Rewrite as $ $(2x^2 + 3)^{-4}$ $ and use chain rule.

13. Differentiate $y = \text{cos}(\text{ln}(x))$.

    $-\frac{\text{sin}(\text{ln}(x))}{x}$. Chain rule: $-\sin(\ln(x)) \times \frac{1}{x}$.

14. Differentiate $y = (\text{e}^x)^2$.

    $2\text{e}^{2x}$. Use power rule: $(\text{e}^x)^2 = \text{e}^{2x}$.

15. Differentiate $y = \text{sin}^4(x)$.

    $4\text{sin}^3(x)\text{cos}(x)$. Power rule: $4\sin^3(x) \times \cos(x)$.

16. Differentiate $y = \text{ln}(x^2 + 2x)$.

    $\frac{2x + 2}{x^2 + 2x}$. Natural log derivative: $\frac{1}{x^2 + 2x} \times (2x + 2)$.

17. What is the derivative of $y = (3x + 2)^4$ using the Chain Rule?

    $12(3x + 2)^3$. Power rule with chain rule: $4(3x + 2)^3 \times 3$.

18. Differentiate $y = \frac{1}{\text{sin}(x^2)}$.

    $-\frac{2x\text{cos}(x^2)}{\text{sin}^2(x^2)}$. Use quotient rule or rewrite as $\csc(x^2)$.

19. Identify the inner function in $y = \text{ln}(\text{cos}(x))$.

    $u = \text{cos}(x)$. The cosine function is inside the natural log.

20. Find the derivative of $y = \frac{1}{\text{e}^{2x}}$.

    $-2\text{e}^{-2x}$. Rewrite as $\text{e}^{-2x}$ and differentiate.

21. Differentiate $y = \text{sin}(\text{e}^x)$.

    $\text{e}^x\text{cos}(\text{e}^x)$. Cosine of $\text{e}^x$ times derivative of $\text{e}^x$.

22. Differentiate $y = \text{e}^{\text{tan}(x)}$.

    $\text{sec}^2(x)\text{e}^{\text{tan}(x)}$. Exponential times derivative of tangent function.

23. Differentiate $y = \text{sin}^3(x)$.

    $3\text{sin}^2(x)\text{cos}(x)$. Power rule: $3\sin^2(x) \times \cos(x)$.

24. Differentiate $y = (5x + 3)^7$.

    $35(5x + 3)^6$. Power rule: $7(5x + 3)^6 \times 5$.

25. Differentiate $y = (\text{ln}(x))^3$.

    $\frac{3(\text{ln}(x))^2}{x}$. Power rule: $3(\ln(x))^2 \times \frac{1}{x}$.

26. Differentiate $y = \text{e}^{\text{cos}(x)}$

    $-\text{sin}(x)\text{e}^{\text{cos}(x)}$. Exponential times derivative of exponent: $-\sin(x)$

27. Differentiate $y = \text{sin}(\text{cos}(x))$.

    $-\text{cos}(\text{cos}(x))\text{sin}(x)$. Composition: sine of cosine times derivative of cosine.

28. Differentiate $y = \tan(4x)$.

    $4\sec^2(4x)$. Tangent derivative is $\sec^2$ times inner derivative.

29. State the formula for the Chain Rule.

    $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Multiplies outer derivative by inner derivative.

30. Differentiate $y = \text{cos}(\text{ln}(x^2))$.

    $-\frac{2\text{sin}(\text{ln}(x^2))}{x}$. Cosine composition with natural log of $x^2$.

31. Differentiate $y = \text{e}^{\text{cos}(x)}$.

    $-\text{sin}(x)\text{e}^{\text{cos}(x)}$. Exponential times derivative of exponent: $-\sin(x)$.

32. Differentiate $y = \text{tan}^3(x)$.

    $3\text{tan}^2(x)\text{sec}^2(x)$. Power rule: $3\tan^2(x) \times \sec^2(x)$.

33. Differentiate $y = \text{tan}(\text{ln}(x))$.

    $\frac{\text{sec}^2(\text{ln}(x))}{x}$. Tangent of natural log: $\sec^2(\ln(x)) \times \frac{1}{x}$.

34. Differentiate $y = \text{cos}(2x^2)$.

    $-4x\text{sin}(2x^2)$. Cosine derivative: $-\sin(2x^2) \times 4x$.

35. Identify the inner function in $y = (4x^2 + 1)^5$.

    $u = 4x^2 + 1$. The expression inside the power is the inner function.

36. Differentiate $y = \text{e}^{\text{e}^x}$.

    $\text{e}^x\text{e}^{\text{e}^x}$. Double exponential: $\text{e}^x \times \text{e}^{\text{e}^x}$.

37. Differentiate $y = \tan^2(x)$.

    $2\tan(x)\sec^2(x)$. Power rule: $2\tan(x) \times \sec^2(x)$.

38. Differentiate $y = \text{ln}(7x^2 + 5)$.

    $\frac{14x}{7x^2 + 5}$. Derivative of $\ln(u)$ is $\frac{1}{u} \times u'$.

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