Exploring Behaviors of Implicit Relations - AP Calculus AB

1. Find $\frac{dy}{dx}$ for $y = \ln(xy)$ implicitly.

   $\frac{dy}{dx} = \frac{1}{x + y}$. From $\frac{dy}{dx} = \frac{1}{xy}(y + x\frac{dy}{dx})$, solve for $\frac{dy}{dx}$.

2. What does $\frac{dy}{dx}$ represent in implicit differentiation?

   The derivative of $y$ with respect to $x$. The rate of change of $y$ with respect to $x$.

3. Differentiate $y^2 = x + x^2y$ implicitly.

   $2y \frac{dy}{dx} = 1 + 2xy + x^2 \frac{dy}{dx}$. Product rule on right side, power rule on left.

4. Differentiate $x + y = e^{xy}$ implicitly.

   $1 + \frac{dy}{dx} = e^{xy}(y + x \frac{dy}{dx})$. Chain rule on exponential function on right side.

5. Differentiate $e^{xy} = x + y$ implicitly with respect to $x$.

   $e^{xy}(y + x \frac{dy}{dx}) = 1 + \frac{dy}{dx}$. Chain rule on left, standard derivatives on right.

6. Differentiate $x^3 + y^3 = 3xy$ implicitly.

   $3x^2 + 3y^2 \frac{dy}{dx} = 3(y + x \frac{dy}{dx})$. Apply product rule to right side.

7. Identify the error: $\frac{d}{dx}(xy) = x \frac{dy}{dx} + y$.

   Correct: $\frac{d}{dx}(xy) = x \frac{dy}{dx} + y \frac{dx}{dx}$. Missing $\frac{dx}{dx} = 1$ in the product rule.

8. Differentiate $\sin(x + y) = y$ implicitly.

   $\cos(x + y)(1 + \frac{dy}{dx}) = \frac{dy}{dx}$. Chain rule on $\sin(x + y)$ gives $\cos(x + y)(1 + \frac{dy}{dx})$.

9. Differentiate $x^4 + y^4 = 16$ implicitly.

   $4x^3 + 4y^3 \frac{dy}{dx} = 0$. Apply power rule: $4x^3 + 4y^3\frac{dy}{dx} = 0$.

10. Differentiate $x = \tan(xy)$ implicitly.

    $1 = \sec^2(xy)(y + x \frac{dy}{dx})$. Derivative of $\tan$ is $\sec^2$, apply chain rule.

11. What is the implicit derivative of $x^3 + y^3 = 6xy$?

    $3x^2 + 3y^2 \frac{dy}{dx} = 6(y + x \frac{dy}{dx})$. Apply product rule to right side: $6(y + x\frac{dy}{dx})$.

12. What is the implicit derivative of $y^3 + 3x^2y = 12$?

    $3y^2 \frac{dy}{dx} + 6xy + 3x^2 \frac{dy}{dx} = 0$. Apply chain rule to $y^3$ and product rule to $3x^2y$.

13. What is the implicit derivative of $y = \sin(xy)$?

    $\frac{dy}{dx} = \cos(xy)(y + x \frac{dy}{dx})$. Chain rule on $\sin(xy)$ equals $\frac{dy}{dx}$.

14. What is the implicit derivative of $y = x + \sin(y)$?

    $\frac{dy}{dx} = 1 + \cos(y) \frac{dy}{dx}$. Chain rule on $\sin(y)$ gives $\cos(y)\frac{dy}{dx}$.

15. What is the purpose of implicit differentiation?

    To find derivatives when not in explicit form. Enables differentiation of relations not solved for $y$.

16. What is implicit differentiation?

    Differentiating equations not solved for one variable in terms of others. Used when $y$ cannot be easily isolated.

17. Differentiate $x^2 + y^2 = 1$ implicitly with respect to $x$.

    $2x + 2y \frac{dy}{dx} = 0$. Apply power rule to both terms, treat $y$ as function of $x$.

18. Differentiate $x^2 - y^2 = 1$ implicitly.

    $2x - 2y \frac{dy}{dx} = 0$. Difference of squares: coefficients have opposite signs.

19. State the Chain Rule for implicit differentiation.

    Differentiate outer, multiply by derivative of inner. Essential for composite functions involving $y$.

20. Differentiate $\tan(xy) = x$ implicitly.

    $\sec^2(xy)(y + x \frac{dy}{dx}) = 1$. Chain rule: derivative of $\tan$ is $\sec^2$.

21. Find $\frac{dy}{dx}$ for $xy + y = 3x$.

    $\frac{dy}{dx} = \frac{3 - y}{x + 1}$. Factor out terms with $y$: $y(x + 1) = 3x$.

22. Find $\frac{dy}{dx}$ for $\cos(xy) = x$.

    $\frac{dy}{dx} = \frac{y\sin(xy) - 1}{x\sin(xy)}$. Chain rule gives $-\sin(xy)(y + x\frac{dy}{dx}) = 1$, solve.

23. Differentiate $x^2 + y^2 = 25$ implicitly.

    $2x + 2y \frac{dy}{dx} = 0$. Same as $x^2 + y^2 = 1$ with different radius.

24. Differentiate $y = x^2 + \tan(y)$ implicitly.

    $\frac{dy}{dx} = 2x + \sec^2(y) \frac{dy}{dx}$. Chain rule on $\tan(y)$ gives $\sec^2(y)\frac{dy}{dx}$.

25. Find $\frac{dy}{dx}$ for $xy = \ln(x)$.

    $\frac{dy}{dx} = \frac{1 - y}{x}$. From product rule: $y + x\frac{dy}{dx} = \frac{1}{x}$.

26. Find $\frac{dy}{dx}$ for $xy = 4$ using implicit differentiation.

    $\frac{dy}{dx} = -\frac{y}{x}$. Use product rule: $y + x\frac{dy}{dx} = 0$, solve for $\frac{dy}{dx}$.

27. What is the implicit differential of $x^2 + xy = 10$?

    $2x + y + x \frac{dy}{dx} = 0$. Product rule on $xy$ term.

28. Find $\frac{dy}{dx}$ for $x^2 - xy + y^2 = 7$.

    $\frac{dy}{dx} = \frac{y - 2x}{2y - x}$. Differentiate each term, collect $\frac{dy}{dx}$ terms, solve.

29. Find $\frac{dy}{dx}$ for $y = \cos(xy)$ implicitly.

    $\frac{dy}{dx} = -\sin(xy)(y + x \frac{dy}{dx})$. Chain rule: $\frac{dy}{dx} = -\sin(xy)(y + x\frac{dy}{dx})$.

30. Differentiate $x = \cos(xy)$ implicitly.

    $1 = -\sin(xy)(y + x \frac{dy}{dx})$. Derivative of $\cos$ is $-\sin$, apply chain rule.

31. Differentiate $x^2y^2 = 1$ implicitly.

    $2xy^2 + 2x^2y \frac{dy}{dx} = 0$. Product rule: $2xy^2 + 2x^2y\frac{dy}{dx} = 0$.

32. Find $\frac{dy}{dx}$ for $x^2 + y^2 = 4$.

    $\frac{dy}{dx} = -\frac{x}{y}$. Circle equation: slope is negative reciprocal of radius slope.

33. What is the implicit differentiation of $x^2y + y^2 = 1$?

    $2xy + x^2 \frac{dy}{dx} + 2y \frac{dy}{dx} = 0$. Product rule on $x^2y$, power rule on $y^2$.

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