Implicitly Defined Functions - AP Precalculus

1. How is implicit differentiation used in related rates?

   To relate rates of change of different variables. Connects how variables change with respect to time.

2. Identify the term to differentiate implicitly: $x^2 + xy + y = 1$.

   $xy$. The product term requires the product rule for differentiation.

3. Differentiate $x^2y + y^3 = 7$ implicitly.

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

4. What does it mean to implicitly differentiate $y^2 = 4x$?

   Find $\frac{dy}{dx}$ for $y^2 = 4x$. Apply chain rule to find the derivative relationship.

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

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

6. What is the chain rule for implicit differentiation?

   Differentiate both sides with respect to $x$, apply $\frac{dy}{dx}$ to $y$ terms. Treat $y$ as a function of $x$ when differentiating.

7. Differentiate $x^2 - 2xy + y^2 = 0$ implicitly.

   $2x - 2y - 2x \frac{dy}{dx} + 2y \frac{dy}{dx} = 0$. Use product rule on $-2xy$ and chain rule on $y^2$.

8. Find $dy/dx$ for $x^3 + y^3 = 6xy$.

   $\frac{2y - 3x^2}{3y^2 - 2x}$. Use implicit differentiation on both sides of the equation.

9. Find the implicit derivative for $x^3 + 3xy + y^3 = 0$.

   $3x^2 + 3y + 3x \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = 0$. Use product rule on $3xy$ and chain rule on cubic terms.

10. Differentiate $x^2 + y = 2xy$ implicitly.

    $2x + \frac{dy}{dx} = 2y + 2x \frac{dy}{dx}$. Differentiate each term and apply product rule to $2xy$.

11. Find $dy/dx$ for the equation $x^2y + y^2 = 10$.

    $-\frac{2xy + 2y}{x^2 + 2y}$. Use product rule and solve for $\frac{dy}{dx}$.

12. What is the implicit derivative for $x^2 + y^2 = 25$?

    $2x + 2y \frac{dy}{dx} = 0$. Standard circle equation differentiated implicitly.

13. What does it mean to differentiate implicitly?

    Differentiate an equation involving multiple variables. Take derivatives without solving for one variable first.

14. Find $dy/dx$ for $x^2 + y^2 = 4xy$.

    $\frac{2x - 4y}{2y - 4x}$. Rearrange after differentiating to solve for $\frac{dy}{dx}$.

15. Differentiate $x^2 + 2xy + y^2 = 0$ implicitly.

    $2x + 2y + 2x \frac{dy}{dx} + 2y \frac{dy}{dx} = 0$. Perfect square form requires product rule on $2xy$.

16. What is the implicit derivative of $e^x + e^y = 1$?

    $e^x + e^y \frac{dy}{dx} = 0$. Apply chain rule to $e^y$ term only.

17. What is the implicit differentiation of $x + y = xy$?

    $1 + \frac{dy}{dx} = y + x \frac{dy}{dx}$. Apply product rule to $xy$ term on right side.

18. Differentiate $xy = 4$ implicitly.

    $y + x \frac{dy}{dx} = 0$. Apply the product rule to the $xy$ term.

19. Differentiate $sin(xy) = 1$ implicitly.

    $cos(xy)(y + x \frac{dy}{dx}) = 0$. Apply chain rule to $\sin(xy)$ using product rule inside.

20. What is the implicit derivative of $x^2 + 2y^2 = 1$?

    $2x + 4y \frac{dy}{dx} = 0$. Chain rule applies to $2y^2$ with coefficient 4.

21. What is an implicitly defined function?

    A function defined by an equation not solved for one variable. The equation contains both variables without solving for one.

22. How do you solve for $dy/dx$ in an implicit equation?

    Use implicit differentiation, solve for $\frac{dy}{dx}$. Collect terms with $\frac{dy}{dx}$ and factor them out.

23. How do you differentiate $xy = 9$ implicitly?

    $y + x \frac{dy}{dx} = 0$. Standard product rule application to $xy$.

24. What is the result of implicit differentiation for $y^2 - x^2 = 1$?

    $2y \frac{dy}{dx} - 2x = 0$. Differentiate each term using appropriate rules.

25. Differentiate $x^2 + y^2 = 9$ implicitly.

    $2x + 2y \frac{dy}{dx} = 0$. Basic circle equation requiring chain rule for $y^2$.

26. Find the implicit derivative for $x^2 + 3xy + 2y^2 = 0$.

    $2x + 3y + 3x \frac{dy}{dx} + 4y \frac{dy}{dx} = 0$. Mixed terms require product rule and chain rule.

27. What is the chain rule in terms of implicit differentiation?

    Use $\frac{dy}{dx}$ for terms of $y$ when differentiating with respect to $x$. Apply $\frac{dy}{dx}$ to all $y$ terms when differentiating.

28. Identify the implicit differentiation of $x^3 = y^3$.

    $3x^2 = 3y^2 \frac{dy}{dx}$. Equal powers differentiate to proportional expressions.

29. What is the derivative of $y$ with respect to $x$ for $x^2 + y^2 = 1$?

    $-\frac{x}{y}$. Apply implicit differentiation to the circle equation.

30. Differentiate $3x + 4y = 12$ implicitly.

    $3 + 4\frac{dy}{dx} = 0$. Simple linear equation with constant coefficients.

31. Differentiate $x^2y - y = 3$ implicitly.

    $2xy + x^2 \frac{dy}{dx} - \frac{dy}{dx} = 0$. Use product rule on $x^2y$ and chain rule on $y$.

32. Find the implicit derivative of $x^2 + xy + y^2 = 7$.

    $2x + y + x \frac{dy}{dx} + 2y \frac{dy}{dx} = 0$. Use product rule on $xy$ and chain rule on $y^2$.

33. What is the implicit derivative of $e^x + y = 5$?

    $e^x + \frac{dy}{dx} = 0$. Simple linear equation with exponential and linear terms.

34. Differentiate $sin(x) + cos(y) = 1$ implicitly.

    $cos(x) - sin(y) \frac{dy}{dx} = 0$. Apply chain rule to trigonometric functions of $x$ and $y$.

35. Identify the implicit function in $x^2 + y^2 = 1$.

    $x^2 + y^2 = 1$. The entire equation is implicit as written.

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