Mr. Integral
Math notes/Calculus I/The Derivative

Implicit and Logarithmic Differentiation — Cheat sheet

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Core rule

Differentiate an implicit relation using y=y(x)y=y(x), collect y′y' and solve only where the coefficient is nonzero.

(ln⁡∣u∣)′=u′/u,(\ln|u|)'=u'/u, (f−1)′(x)=1/f′(f−1(x)).(f^{-1})'(x)=1/f'(f^{-1}(x)).

Watch for

A variable exponent needs logarithmic differentiation or a suitable rewrite. An inverse must have a chosen branch, and its derivative denominator must be nonzero.