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In mathematics, the Clarke generalized derivatives are types generalized of derivatives that allow for the differentiation of nonsmooth functions. The Clarke derivatives were introduced by Francis Clarke in 1975.
In this case one needs to consider the Fréchet derivative or Gateaux derivative. Example The five terms in the following expression correspond in the obvious way to the five partitions of the set { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} , and in each case the order of the derivative of f {\displaystyle f} is the number of parts in the partition:
velocity is the derivative (with respect to time) of an object's displacement (distance from the original position) acceleration is the derivative (with respect to time) of an object's velocity, that is, the second derivative (with respect to time) of an object's position. For example, if an object's position on a line is given by
The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances.
A particular solution is derived from the general solution by setting the constants to particular values, often chosen to fulfill set 'initial conditions or boundary conditions'. [22] A singular solution is a solution that cannot be obtained by assigning definite values to the arbitrary constants in the general solution.
The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on R n. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold
Bass diffusion model; Black–Scholes equation; Economic growth. Solow–Swan model ′ = [()] Ramsey–Cass–Koopmans model; Dynamic stochastic general equilibrium [8]; Feynman–Kac formula
Consider the following second-order problem, ′ + + = () =, where = {,, <is the Heaviside step function.The Laplace transform is defined by, = {()} = ().Upon taking term-by-term Laplace transforms, and utilising the rules for derivatives and integrals, the integro-differential equation is converted into the following algebraic equation,