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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.
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
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.
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): ... Its partial derivatives are
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
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.
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 ...
This order of things puts everything in the direct line of fire of the chain rule; the partial derivatives , and are easy to compute and at the end, the original equation stands ready for immediate use."