Vault Of Euclid

Implicit Differentiation via Partial Derivatives

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Implicit Differentiation Given the implicitly defined function $\sin (x^2 y^2) + y^3 = x + y$, find $y'$. Note: this is the same problem as given in Example 2.6.4 of Section 2.6, where the solution took about a full page to find.
implicit-differentiationpartial-derivativesmultivariable-calculuschain-ruleimplicit-function-theoremcalculus-iiievaluate-derivativetrigonometric-functionsfunctions-of-two-variables

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What this problem tests

Tests the ability to compute partial derivatives and apply the Implicit Function Theorem formula.

This problem demonstrates a powerful shortcut taught in multivariable calculus for finding derivatives of implicit functions, bypassing the tedious algebra of single-variable implicit differentiation.

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