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  2. Chain rule - Wikipedia

    en.wikipedia.org/wiki/Chain_rule

    In this situation, the chain rule represents the fact that the derivative of f ∘ g is the composite of the derivative of f and the derivative of g. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula. The chain rule is also valid for Fréchet derivatives in Banach spaces.

  3. Related rates - Wikipedia

    en.wikipedia.org/wiki/Related_rates

    Big idea: use chain rule to compute rate of change of distance between two vehicles. Plan: Choose coordinate system; Identify variables; Draw picture; Big idea: use chain rule to compute rate of change of distance between two vehicles; Express c in terms of x and y via Pythagorean theorem; Express dc/dt using chain rule in terms of dx/dt and dy/dt

  4. Total derivative - Wikipedia

    en.wikipedia.org/wiki/Total_derivative

    The chain rule has a particularly elegant statement in terms of total derivatives. It says that, for two functions f {\displaystyle f} and g {\displaystyle g} , the total derivative of the composite function f ∘ g {\displaystyle f\circ g} at a {\displaystyle a} satisfies

  5. Matrix calculus - Wikipedia

    en.wikipedia.org/wiki/Matrix_calculus

    In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.

  6. Triple product rule - Wikipedia

    en.wikipedia.org/wiki/Triple_product_rule

    Suppose a function f(x, y, z) = 0, where x, y, and z are functions of each other. Write the total differentials of the variables = + = + Substitute dy into dx = [() + ()] + By using the chain rule one can show the coefficient of dx on the right hand side is equal to one, thus the coefficient of dz must be zero () + = Subtracting the second term and multiplying by its inverse gives the triple ...

  7. Vector calculus identities - Wikipedia

    en.wikipedia.org/wiki/Vector_calculus_identities

    2.5 Chain rule. 2.6 Dot product rule. 2.7 Cross product rule. 3 Second derivative identities. ... We have the following special cases of the multi-variable chain rule.

  8. Change of variables (PDE) - Wikipedia

    en.wikipedia.org/wiki/Change_of_variables_(PDE)

    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."

  9. Integration by substitution - Wikipedia

    en.wikipedia.org/wiki/Integration_by_substitution

    In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation , and can loosely be thought of as using the chain rule "backwards."