Implicit Differentiation - AP Calculus AB

1. Differentiate $\frac{x}{y} = 1$ with respect to $x$.

   $\frac{y - x\frac{dy}{dx}}{y^2} = 0$. Apply quotient rule: $\frac{d}{dx}[\frac{x}{y}] = \frac{y - x\frac{dy}{dx}}{y^2}$.

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

   The derivative of $y$ with respect to $x$. It's the rate of change of $y$ with respect to $x$.

3. Differentiate $x^2 - y^2 = 4$ using implicit differentiation.

   $2x - 2y\frac{dy}{dx} = 0$. Apply power rule to both $x^2$ and $y^2$ terms.

4. What is the first step in implicit differentiation?

   Differentiate both sides with respect to $x$. This sets up the differentiation process.

5. Find the derivative of $xy = x + y$ using implicit differentiation.

   $y + x\frac{dy}{dx} = 1 + \frac{dy}{dx}$. Apply product rule to left side.

6. Differentiate $3x^2 + 4y^2 = 12$ with respect to $x$.

   $6x + 8y\frac{dy}{dx} = 0$. Apply power rule to each term.

7. Differentiate $x^2y + y^3 = 0$ using implicit differentiation.

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

8. Differentiate: $x^2 + y^2 = 1$ with respect to $x$.

   $2x + 2y\frac{dy}{dx} = 0$. Apply power rule to each term, using chain rule for $y^2$.

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

   $\frac{dy}{dx} = 2xy + x^2\frac{dy}{dx} + 1$. Use product rule for $x^2y$ term.

10. Differentiate: $\frac{dy}{dx}$ if $x^2 + y^2 = r^2$.

    $2x + 2y \frac{dy}{dx} = 0$. Same as differentiating $x^2 + y^2 = 1$ but with constant $r^2$.

11. Differentiate $x^3 + 3xy + y^3 = 0$ using implicit differentiation.

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

12. Differentiate $2x^2y = 3$ using implicit differentiation.

    $4xy + 2x^2\frac{dy}{dx} = 0$. Use product rule on $2x^2y$.

13. Differentiate $x^2 + y^2 = 9$ using implicit differentiation.

    $2x + 2y\frac{dy}{dx} = 0$. Same form as circle equation with radius 3.

14. Differentiate $x^2 + 2xy + y^2 = 16$ with respect to $x$.

    $2x + 2y + 2x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$. This is $(x+y)^2 = 16$; use chain rule.

15. What is the result of differentiating $x + y = 1$ implicitly?

    $1 + \frac{dy}{dx} = 0$. Each term differentiates to its coefficient.

16. Find $\frac{dy}{dx}$ for $y^3 + xy = 1$.

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

17. Differentiate $xy^2 = 4$ with respect to $x$.

    $y^2 + 2xy\frac{dy}{dx} = 0$. Apply product rule to $xy^2$.

18. Find the derivative of $y$ in $3x^2 + 2xy - y^3 = 0$.

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

19. Which rule is applied when differentiating $y^2$ with respect to $x$?

    The Chain Rule. Since $y$ depends on $x$, we need $\frac{dy}{dx}$ when differentiating $y^2$.

20. Differentiate $x^3 + y^3 = 6xy$ using implicit differentiation.

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

21. What is implicit differentiation?

    A technique to find derivatives of equations not solved for $y$. Used when $y$ is not isolated on one side of the equation.

22. Differentiate $\frac{x^3}{y} = 1$ with respect to $x$.

    $3x^2 - \frac{x^3\frac{dy}{dx}}{y^2} = 0$. Apply quotient rule to $\frac{x^3}{y}$.

23. What is the derivative of $xy = 1$ using implicit differentiation?

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

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

    $2x + y + x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$. Apply product rule to $xy$ and power rule to other terms.

25. Differentiate $x^2y^2 = 4$ with respect to $x$.

    $2xy^2 + 2x^2y\frac{dy}{dx} = 0$. Use product rule: $\frac{d}{dx}[x^2y^2] = 2x(y^2) + x^2(2y\frac{dy}{dx})$.

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

    $2y\frac{dy}{dx} = 2x$. Apply power rule to $y^2$ using chain rule.

27. Find the derivative of $x^2y + y^2 = 10$ using implicit differentiation.

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

28. Differentiate $x + y = xy$ with respect to $x$.

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

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