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In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. [1] [2] [3] Let () = (), where both f and g are differentiable and ()
(The chain rule) d ( G ∘ F ) ( u ; x ) = d G ( F ( u ) ; d F ( u ; x ) ) {\displaystyle d(G\circ F)(u;x)=dG(F(u);dF(u;x))} for all u ∈ U {\displaystyle u\in U} and x ∈ X . {\displaystyle x\in X.} (Importantly, as with simple partial derivatives , the Gateaux derivative does not satisfy the chain rule if the derivative is permitted to be ...
In a finite partial order (or more generally a partial order satisfying the ascending chain condition) all lower sets have this form. The union of any two lower sets is another lower set, and the union operation corresponds in this way to a join operation on antichains: A ∨ B = { x ∈ A ∪ B : ∄ y ∈ A ∪ B such that x < y ...
Difficult integrals may often be evaluated by changing variables; this is enabled by the substitution rule and is analogous to the use of the chain rule above. Difficult integrals may also be solved by simplifying the integral using a change of variables given by the corresponding Jacobian matrix and determinant . [ 1 ]
chain rule: If g is differentiable at p and h is differentiable at g(p), then ... Instead of building the directional derivative using partial derivatives, ...
The Laplace operator or Laplacian on R 3 is a second-order partial differential operator Δ given by the divergence of the gradient of a scalar function of three variables, or explicitly as = + +. Analogous operators can be defined for functions of any number of variables.
Chain rule – For derivatives of composed functions; Difference quotient – Expression in calculus; Differentiation of integrals – Problem in mathematics; Differentiation rules – Rules for computing derivatives of functions; General Leibniz rule – Generalization of the product rule in calculus
To start, we should choose a working definition of the value of () =, where is any real number. Although it is feasible to define the value as the limit of a sequence of rational powers that approach the irrational power whenever we encounter such a power, or as the least upper bound of a set of rational powers less than the given power, this type of definition is not amenable to differentiation.