Implicit Differentiation – Quick Review

Key Idea: Implicit Differentiation

• Used when y is not isolated (relation, not explicit function).
• Treat y as an unknown function of x. Whenever you differentiate a term containing y, multiply by \frac{dy}{dx} (chain rule).
• Apply familiar rules (power, product, quotient, chain) exactly as for explicit functions.
• After differentiating both sides, algebraically solve for \frac{dy}{dx}.

Core Procedure

• Differentiate both sides of the given equation with respect to x.
• Attach \frac{dy}{dx} to every y–derivative.
• Use product/chain rule wherever x and y are multiplied or nested.
• Collect all \frac{dy}{dx} terms on one side.
• Factor out \frac{dy}{dx}.
• Isolate by dividing: \frac{dy}{dx}=\dfrac{\text{terms without }\frac{dy}{dx}}{\text{coefficient of }\frac{dy}{dx}}.

Quick Reference: Product & Chain Rules

• Product rule: (uv)' = u'v + uv'.
• Chain rule with y: If f(y), then \dfrac{d}{dx}f(y)=f'(y)\frac{dy}{dx}.

Worked-Out Results (for rapid recall)

• x^2 = 3y^4 + x \;\Rightarrow\; \displaystyle \frac{dy}{dx}=\frac{2x-1}{12y^3}
• e^{2x}=x^3y^4 \;\Rightarrow\; \displaystyle \frac{dy}{dx}=\frac{2e^{2x}-3x^2y^4}{4x^3y^3}
• x^5+3x^2y^3+y^5=2 \;\Rightarrow\; \displaystyle \frac{dy}{dx}=\frac{-5x^4-6xy^3}{9x^2y^2+5y^4}
• 3x^3y^4-\ln(5x^3y)=5 \;\Rightarrow\; \displaystyle \frac{dy}{dx}=\frac{15x^2y-45x^3y^5}{60x^3y^3+5x^2}

Common Pitfalls

• Forgetting the extra \frac{dy}{dx} factor when differentiating any y term.
• Skipping the product rule when x and y appear in the same factor (e.g., x^3y^4).
• Failing to distribute negative signs or combine like terms before factoring \frac{dy}{dx}.

Rapid Checklist Before Finishing

• [ ] Did every y-derivative get a \frac{dy}{dx}?
• [ ] Were product/chain rules correctly applied?
• [ ] Are all \frac{dy}{dx} terms on one side and factored?
• [ ] Final answer solved explicitly for \frac{dy}{dx}?