AP Precalculus Flashcards: Implicitly Defined Functions

What this deck covers

This deck focuses on Implicitly Defined Functions, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: How is implicit differentiation used in related rates?

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

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

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

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

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

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

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

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

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

Flashcard 6: What is the chain rule for implicit differentiation?

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 13: What does it mean to differentiate implicitly?

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

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

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

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

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

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

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

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

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

Flashcard 18: Differentiate $xy = 4$ implicitly.

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

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

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

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

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

Flashcard 21: What is an implicitly defined function?

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

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

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

Flashcard 23: How do you differentiate $xy = 9$ implicitly?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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