Differential calculus chain rule

A fifth formula, for use in situations in which a new variable X(P,T) is to be introduced, is an example of the chain rule of differential calculus. The formula is... 

The chain rule is an indispensable tool in differential calculus. It allows for the simplification of derivatives of composite functions. 

Since multilayer perceptions use neurons that have differentiable functions, it was possible, using the chain rule of calculus, to derive a delta rule for training similar in form and function to that for perceptions. The result of this clever mathematics is a powerful and relatively efficient iterative method for multilayer perceptions. The rule for changing weights into a neuron unit becomes... 

Finally, the partial derivative dVfdx may be evaluated from differential expressions such as (2.6) using the chain rule of elementary calculus. From (2.5), F is a function of T and P, or V = V(T, P). Let T and P each be functions of two other variables X and y,... 

The substitution rule is the reverse of the chain rule of differential calculus. The method looks more complicated than it actually is. Let us consider an example We are looking for an antiderivative of... 

We have the mathematical proposition that the covariant differentiation (24)-(26) defines tensor fields of the corresponding index picture. Its proof follows the simple mathematical rule of thumb Tensor calculus is an application of the chain rule. With eq.(8), we have... 

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