Cover's theorem

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):

$\mathrm {P} (X_{1}=x_{1},\ldots ,X_{n}=x_{n})=\prod _{i=1}^{n}\mathrm {P} (X_{i}=x_{i}\mid X_{i+1}=x_{i+1},\ldots ,X_{n}=x_{n})$

• Chain rule for Kolmogorov complexity:

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

• Chain rule for information entropy:

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