Скачать или смотреть Calculus 1 — 15.2: Implicit Differentiation

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

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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.