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In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g.More precisely, if = is the function such that () = (()) for every x, then the chain rule is, in Lagrange's notation, ′ = ′ (()) ′ (). or, equivalently, ′ = ′ = (′) ′.
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 ...
If the direction of derivative is not repeated, it is called a mixed partial derivative. If all mixed second order partial derivatives are continuous at a point (or on a set), f is termed a C 2 function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem:
The matrix derivative is a convenient notation for keeping track of partial derivatives for doing calculations. The Fréchet derivative is the standard way in the setting of functional analysis to take derivatives with respect to vectors. In the case that a matrix function of a matrix is Fréchet differentiable, the two derivatives will agree ...
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 second line is obtained using the total derivative, where ∂f /∂∇ρ is a derivative of a scalar with respect to a vector. [ Note 4 ] The third line was obtained by use of a product rule for divergence .
3.3 Proof using the reciprocal rule or chain rule. ... Partial derivative; ... the quotient rule is a method of finding the derivative of a function that is the ratio ...