Chain Rule - AP Calculus BC

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

   $\sec^2(x) e^{\tan(x)}$. Chain rule: $e^u \times u'$ where $u = \tan(x)$

2. Differentiate $f(x) = \ln(e^x + 1)$.

   $\frac{e^x}{e^x + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = e^x + 1$.

3. Identify $u$ in $y = (\text{ln}(x))^4$.

   $u = \text{ln}(x)$. The natural logarithm function raised to power 4.

4. Identify the inner function in $f(g(x)) = \tan(\frac{x^2}{3})$.

   $g(x) = \frac{x^2}{3}$. The expression inside the tangent function.

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

   $\frac{2x\text{cos}(\text{ln}(x^2))}{x^2}$. Chain rule twice: $\cos(\ln(x^2)) \times \frac{2x}{x^2}$.

6. Differentiate $g(x) = \text{ln}(4x^2 + 1)$.

   $\frac{8x}{4x^2 + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = 4x^2 + 1$.

7. Find $f'(x)$ if $f(x) = \text{sin}(x^3)$.

   $3x^2 \text{cos}(x^3)$. Chain rule: $\cos(u) \times u'$ where $u = x^3$.

8. Find $d/dx$ of $y = \text{cos}(\text{e}^{2x})$.

   $-2\text{e}^{2x}\text{sin}(\text{e}^{2x})$. Chain rule: $-\sin(u) \times u'$ where $u = e^{2x}$.

9. Identify $u$ in $h(x) = (3x^2 + 2x)^5$.

   $u = 3x^2 + 2x$. The expression being raised to the 5th power.

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

    $\frac{1}{x}$. Simplifies to $x$ since $e^{\ln(x)} = x$.

11. Differentiate $e^{x^2}$ using the Chain Rule.

    $2xe^{x^2}$. Chain rule: derivative of $e^u$ is $e^u$ times $u'$.

12. Differentiate $y = \text{e}^{3x+7}$.

    $3\text{e}^{3x+7}$. Chain rule: $e^u \times u'$ where $u = 3x + 7$.

13. Find $dy/dx$ if $y = \text{ln}(\text{e}^{x^2})$.

    $2x$. Simplifies to $x^2$ since $\ln$ and $e$ cancel.

14. Identify $u$ in $y = \text{tan}^2(3x)$.

    $u = \text{tan}(3x)$. The squared tangent of $3x$.

15. Differentiate $f(x) = \text{cos}^3(x)$.

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

16. Find $g'(x)$ if $g(x) = \text{tan}^{-1}(2x^3)$.

    $\frac{6x^2}{1 + 4x^6}$. Chain rule: $\frac{1}{1+u^2} \times u'$ where $u = 2x^3$.

17. Find $y'$ if $y = \text{tan}(\text{ln}(x))$.

    $\frac{1}{x\text{cos}^2(\text{ln}(x))}$. Chain rule twice: $\sec^2(\ln(x)) \times \frac{1}{x}$.

18. Differentiate $y = \text{tan}^2(3x)$ using the Chain Rule.

    $6\text{tan}(3x)\text{sec}^2(3x)$. Chain rule: $2u \times u'$ where $u = \tan(3x)$.

19. Differentiate $f(x) = \text{e}^{\text{sin}(x)}$.

    $\text{cos}(x)\text{e}^{\text{sin}(x)}$. Chain rule: $e^u \times u'$ where $u = \sin(x)$.

20. Differentiate $y = (\text{ln}(x))^4$ using the Chain Rule.

    $\frac{4(\text{ln}(x))^3}{x}$. Power rule with chain rule: $4u^3 \times u'$.

21. Find $f'(x)$ if $f(x) = \text{cos}(\text{e}^x)$.

    $-\text{e}^x\text{sin}(\text{e}^x)$. Chain rule: $-\sin(u) \times u'$ where $u = e^x$.

22. Differentiate $y = (2x + 5)^6$.

    $12(2x+5)^5$. Power rule with chain rule: $6u^5 \times u'$.

23. Which function is the outer function in $f(g(x)) = \text{e}^{3x+7}$?

    $f(u) = \text{e}^u$. The exponential function wraps the linear expression.

24. What is $y'$ if $y = \text{cos}(5x)$?

    $-5\text{sin}(5x)$. Chain rule: $-\sin(u) \times u'$ where $u = 5x$.

25. Differentiate $y = \text{cos}^{-1}(5x)$.

    $-\frac{5}{\text{sqrt}(1 - 25x^2)}$. Chain rule: $-\frac{1}{\sqrt{1-u^2}} \times u'$ where $u = 5x$.

26. Evaluate $\frac{d}{dx}[\text{ln}(x^2 + 1)]$.

    $\frac{2x}{x^2 + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = x^2 + 1$.

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

    $f'(g(x)) \times g'(x)$. Chain rule applied to general composite function $f(g(x))$.

28. Differentiate $f(x) = \text{ln}(\text{e}^x + 1)$.

    $\frac{\text{e}^x}{\text{e}^x + 1}$. Chain rule: $\frac{1}{u} \times u'$ where $u = e^x + 1$.

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

    $\text{sec}^2(x)\text{e}^{\text{tan}(x)}$. Chain rule: $e^u \times u'$ where $u = \tan(x)$.

30. Differentiate $y = \tan^2(3x)$ using the Chain Rule.

    $6\tan(3x)\sec^2(3x)$. Chain rule: $2u \times u'$ where $u = \tan(3x)$.

31. State the formula for the Chain Rule.

    $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Fundamental chain rule formula for composite functions.

32. Identify the outer function in $f(g(x)) = \tan(\frac{x^2}{3})$.

    $f(u) = \tan(u)$. The tangent function wraps the inner expression.

33. Find the derivative of $\text{sin}^2(x)$.

    $2\text{sin}(x)\text{cos}(x)$. Chain rule: $2\sin(x) \times \cos(x)$ or $\sin(2x)$.

34. Differentiate $f(x) = \text{e}^{x^3}$.

    $3x^2\text{e}^{x^3}$. Chain rule: $e^u \times u'$ where $u = x^3$.

35. Which function is the inner function in $f(g(x)) = \text{e}^{3x+7}$?

    $g(x) = 3x+7$. The linear expression in the exponent.

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