Chain rule
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:
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
The chain rule may be written, in Leibniz's 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,
The two versions of the chain rule are related, if and , then
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