Chain rule (disambiguation)

Chain rule may refer to:

• Chain rule in calculus:

$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\cdot \frac{\mathrm{d}u}{\mathrm{d}x}.$

• Cyclic chain rule, or triple product rule:

${\left(\frac{\partial x}{\partial y}\right)}_z{\left(\frac{\partial y}{\partial z}\right)}_x{\left(\frac{\partial z}{\partial x}\right)}_y=-1.$

• Chain rule (probability):

$\mathrm{P}(X_1=x_1,\dots ,X_n=x_n)=\prod _{i=1}^n\mathrm{P}(X_i=x_i\mid X_{i+1}=x_{i+1},\dots ,X_n=x_n)$

• Chain rule for Kolmogorov complexity:

$K(X,Y)=K(X)+K(Y|X)+O(\log (K(X,Y)))$

• Chain rule for information entropy:

$H(X,Y)=H(X)+H(Y|X)$