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Implicit differentiation lets you find slopes and higher derivatives even when y is given indirectly by an equation F(x,y)=0.
The Hessian matrix collects all second-order partial derivatives of a scalar function and measures local curvature.
The Jacobian matrix collects all first-order partial derivatives of a vector-valued function, describing how small input changes linearly affect each output component.
The multivariable chain rule explains how rates of change pass through a pipeline of functions by multiplying the right derivatives (Jacobians) in the right order.
Partial derivatives measure how a multivariable function changes when you wiggle just one input while keeping the others fixed.