In calculus, the chain rule is a formula for computing the derivative of the composition

of two or more functions.

That is, if f and g are functions, then the chain rule expresses the derivative of their

composition f ∘ g {the function which maps x to f(g(x)) } in terms of the derivatives of f

and g and the product of functions as follows:

    ( f ∘ g )^T = ( f^T ∘ g) * g^T

This can be written more explicitly in terms of the variable.

Let F = f ∘ g, or equivalently, F(x) = f(g(x)) for all x. Then one can also write

    F^T (x) = f^T {g( x)} * g^T(x)       [Here f^T = df/dx, same for other places also]

The chain rule may be written in Leibniz notation in the following way.

If a variable z depends on the variable y, which itself depends on the variable x, so

that y and z are therefore dependent variables, then z, via the intermediate variable

of y, depends on x as well. The chain rule then states,

    dz/dx = (dz/dy) * (dy/dx)

The two versions of the chain rule are related,

If z = f(y) and y = g(x), then

dz/dx = (dz/dy) * (dy/dx) = f^T (y) * g^T (x) = f^T {g(x)} * g^T (x)