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Mathematics

Implicit Differentiation and the Chain Rule in Higher Dimensions

Quick fact

In multivariable calculus, implicit differentiation works because the total derivative of the equation F(x,y)=0 with respect to x is zero. By the chain rule, this yields dy/dx = -Fx / Fy, a formula that works even when y cannot be explicitly solved for.

Why this is interesting

You know that a circle is x² + y² = 1, but can you find dy/dx without solving for y? How do you handle a whole surface like F(x,y,z)=0?