In calculus, the chain rule describes the derivative of a "function of a function": the composition of two function, where the output z is a given function of an intermediate variable y which is in turn a given function of the input variable x.
Suppose that y is given as a function and that z is given as a function . The rate at which z varies in terms of y is given by the derivative , and the rate at which y varies in terms of x is given by the derivative . So the rate at which z varies in terms of x is the product , and substituting we have the chain rule
In order to convert this to the traditional (Leibniz) notation, we notice
In mnemonic form the latter expression is
which is easy to remember, because it as if dy in the numerator and the denominator of the right hand side cancels.
The extension of the chain rule to multivariable functions may be achieved by considering the derivative as a linear approximation to a differentiable function.
Now let and be functions with F having derivative DF at and G having derivative DG at . Thus DF is a linear map from and DG is a linear map from . Then is differentiable at with derivative
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