Why some equations are too complex to solve for y first
In short: When a relationship between x and y is tangled in an implicit equation, standard differentiation fails. This video explains how to find second derivatives by differentiating both sides of the expression simultaneously.
Implicit differentiation is essential when variables cannot be easily isolated. Instead of solving for y first—which may be algebraically impossible or messy—you differentiate the entire equation with respect to x, treating y as a function of x. This requires applying the chain rule whenever y appears.
The process extends to finding second derivatives (y''). By differentiating the first derivative expression again, you can determine the curvature and rate of change for complex curves like $x^2 + y^3 = 4$ or trigonometric identities involving both variables.
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Sources
- Summary by Intellect · source: Second derivatives (implicit equations): find expression | AP Calculus AB | Khan Academy
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