Unit 3.2 – Implicit Differentiation

AP® Calculus AB & BC | Derivatives When \( y \) is “Trapped”

Implicit Differentiation lets you find derivatives for equations where \( y \) isn’t isolated as a function of \( x \). It’s required whenever you see equations like circles, ellipses, or formulas mixing \( x \) and \( y \) together. On the AP® Exam, all related rates and tangent line questions for implicitly defined curves use this strategy!

🔗 Implicit Differentiation: Main Formula & Process

Rule for Derivative Terms Involving \(y\):

When differentiating any term with \(y\), multiply by \(\frac{dy}{dx}\):

\[ \frac{d}{dx}[y^n] = n y^{n-1} \frac{dy}{dx} \]

In general, for \(F(x,y)\) with \(y\) as a function of \(x\):
Differentiate both sides with respect to \(x\), treating \(y\) as \(y(x)\).

Step-by-step process:

Differentiation BOTH sides of the equation with respect to \(x\).
For every term with \(y\), use Chain Rule: tack on \(\frac{dy}{dx}\).
Solve for \(\frac{dy}{dx}\) (collect terms, isolate, factor if needed).

📑 Typical AP® Chain Rule Forms (Recap)

\(\frac{d}{dx}[\sin(y)] = \cos(y) \frac{dy}{dx}\)
\(\frac{d}{dx}[\ln(y)] = \frac{1}{y} \frac{dy}{dx}\)
\(\frac{d}{dx}[e^{y}] = e^{y} \frac{dy}{dx}\)
\(\frac{d}{dx}[y^n] = n y^{n-1} \frac{dy}{dx}\)

Any function of y differentiated with respect to x gets multiplied by \(\frac{dy}{dx}\)!

📖 Worked Examples

Example 1: Circle equation

Differentiate both sides:
\(x^2 + y^2 = 9\)
\(2x + 2y \frac{dy}{dx} = 0\)
\(2y \frac{dy}{dx} = -2x \implies \frac{dy}{dx} = -\frac{x}{y}\)

Example 2: Equation with mixed xy

Differentiate both sides:
\(x^2 y + x = 7\)
Product Rule for \(x^2 y\): \(2x y + x^2 \frac{dy}{dx} + 1 = 0\)
\(x^2 \frac{dy}{dx} = -2x y - 1\)
\(\displaystyle \frac{dy}{dx} = \frac{-2x y - 1}{x^2}\)

Example 3: Higher power in y

\(y^3 + x^2 = 5y\)
\(3y^2\frac{dy}{dx} + 2x = 5\frac{dy}{dx}\)
\(3y^2\frac{dy}{dx} - 5\frac{dy}{dx} = -2x\)
\(\frac{dy}{dx}(3y^2 - 5) = -2x\)
\(\frac{dy}{dx} = \frac{-2x}{3y^2 - 5}\)

💡 Tips, Tricks, & Shortcuts

Always multiply by \(\frac{dy}{dx}\) for every \(y\) term!
Encapsulate with Chain Rule: treat every \(y\) as \(y(x)\).
When you see \(xy\), use the Product Rule.
For “find tangent line,” first get \(\frac{dy}{dx}\), then use point values.
If you get a derivative with both \(x\) and \(y\), plug in points as needed to find slope.

✨ Short Notes & Memory Tricks

Say: “Every \(y\) gets a \(\frac{dy}{dx}\)!”
If stuck, solve for \(\frac{dy}{dx}\) by factoring or grouping.
Don't forget the Product Rule for all \(xy\), \(x^2y\), etc.
If the equation involves trigonometric or logarithm of \(y\), use the chain rule (multiply by \(\frac{dy}{dx}\)).

📝 Practice Problems

Try These Yourself:

\(x^2 + y^2 = 16\)
\(xy + y^2 = 10\)
\(e^y + x = y\)
\(\sin y + x^3 = y^2\)

Answers:

\(\frac{dy}{dx} = -\frac{x}{y}\)
\(\frac{dy}{dx} = \frac{-x}{y + 2x}\)
\(\frac{dy}{dx} = \frac{1}{1 - e^y}\)
\(\frac{dy}{dx} = \frac{-3x^2}{\cos y + 2y}\)

✏️ AP® Exam Success – Implicit Differentiation Tips

Write \(\frac{dy}{dx}\) for every \(y\) derivative step—AP® graders must see it.
Always collect all terms with \(\frac{dy}{dx}\) on one side before solving for it.
For tangent lines: compute \(\frac{dy}{dx}\) formula, then plug the specific \(x\), \(y\).
Watch for chain rule in trigonometric and exponential terms involving \(y\)!
Box your final answer for clarity in FRQs.