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Implicit Differentiation: Three Steps That Always Work

When an equation tangles x and y together so tightly that you can't solve for y, how do you find the slope? Implicit differentiation handles it with a surprisingly mechanical three-step procedure: differentiate both sides, apply the chain rule to every y-term, then solve algebraically for dy/dx. This video walks through the method from motivation to verification. Key concepts covered: • Why equations like x²y³ + sin(xy) = 7 can't be differentiated explicitly • The chain rule insight: differentiating y³ produces 3y²(dy/dx), not just 3y² • The three-step method: differentiate → chain rule on y-terms → collect, factor, divide • Example 1: x³ + y³ = 5, with full verification against the explicit solution • Example 2: 3y² + sin(y) = 4x⁵, where no explicit formula for y exists • Why implicit derivatives naturally contain both x and y • The most common mistake: forgetting to attach dy/dx when differentiating y-terms — ORIGINAL SOURCE This video is based on content from:    • Calculus 1 Lecture 2.7:  Implicit Differen...   All credit for the original instruction and exposition belongs to the original creator. — About Ludium Learn. Play. Discover. Ludium distills long lectures into focused concept videos so you can learn core ideas efficiently. GitHub: https://github.com/Augustinus12835/au... #ImplicitDifferentiation #Calculus #ChainRule #Derivatives #MathEducation #Ludium