For composite functions, multiply the outer derivative by the inner derivative. Essential for differentiating nested functions.
Physics — Velocity of projectile: d/dt(height(time)) requires chain rule.
Machine learning — Backpropagation in neural networks is chain rule applied.
Economics — Elasticity calculations: (dQ/dP)(P/Q).
Biology — Population change through dependent factors.
Engineering — Sensitivity analysis in cascaded systems.
Finance — Option Greeks like Delta and Gamma.
Chemistry — Related rates in chemical kinetics.
Robotics — Jacobian matrix calculations.
Signal processing — Filter cascade analysis.
Computer graphics — Hierarchical transformations.
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