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The chain rule for total derivatives implies a chain rule for partial derivatives. Recall that when the total derivative exists, the partial derivative in the i -th coordinate direction is found by multiplying the Jacobian matrix by the i -th basis vector.
The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables.
If the direction of derivative is not repeated, it is called a mixed partial derivative. ... Triple product rule, also known as the cyclic chain rule. Notes
By applying the chain rule repeatedly to these operations, partial derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor of more arithmetic operations than the original program.
The partial derivative of f with respect to x does not give the true rate of change of f with respect to changing x because changing x necessarily changes y. However, the chain rule for the total derivative takes such dependencies into account. Write () = (, ()). Then, the chain rule says
The chain rule applies in some of the cases, but unfortunately does not apply in matrix-by-scalar derivatives or scalar-by-matrix derivatives (in the latter case, mostly involving the trace operator applied to matrices). In the latter case, the product rule can't quite be applied directly, either, but the equivalent can be done with a bit more ...
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators [1]), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives ...