AP Calculus AB Unit 3.2: Implicit Differentiation

Practice AP Calculus Unit 3.2 questions on differentiating implicit curves and using the result to analyze tangent lines and rates.

Covered syllabus points

[Maximum number: 2]

Consider the curve G defined by the equation $y^{3}-y^{2}-y+\frac{1}{4} x^{2}=0$ .

Show that $\frac{d y}{d x}=\frac{-x}{2\left(3 y^{2}-2 y-1\right)}$ .


$\begin{aligned} & \frac{d}{d x}\left(y^{3}-y^{2}-y+\frac{1}{4} x^{2}\right)=0 \\ & \Rightarrow 3 y^{2} \frac{d y}{d x}-2 y \frac{d y}{d x}-\frac{d y}{d x}+\frac{x}{2}=0 \end{aligned}$	Implicit differentiation	Point 1 (P1)
$\begin{aligned} & \Rightarrow\left(3 y^{2}-2 y-1\right) \frac{d y}{d x}=-\frac{x}{2} \\ & \Rightarrow \frac{d y}{d x}=\frac{-x}{2\left(3 y^{2}-2 y-1\right)} \end{aligned}$	Verification	Point 2 (P2)
Scoring Notes for Part A
- P1 is earned only for the correct implicit differentiation of $y^{3}-y^{2}-y+\frac{1}{4}x^{2}=0$ . Alternative notation such as $y^{\prime}$ is acceptable. - It is sufficient to present $\left(3 y^{2}-2 y-1\right)\frac{d y}{d x}=-\frac{x}{2}$ to earn P2, provided there are no subsequent errors.

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