Derivatives of function

Global derivatives of functions can now be related to the locally defined finite element approximation, given by Equation (2.17), as... 

This means that the derivative of function 1/3 in any direction / is zero... 

It should be evident that the expressions for the Laplace transforms of derivatives of functions can facilitate the solution of differential equations. A trivial example is that of the classical harmonic oscillator. Its equation of motion is given by Eq. (5-33), namely,... 

A A data list in Fig. 9b fu Partial derivative of function / with... 

Alkoxy compounds are commonly called ethers. In current CAS nomenclature they are named as derivatives of functional parent compounds, hydrocarbons, etc., by use of oxy radicals. 

In this chapter we are concerned with equations containing derivatives of functions. Such equations are termed differential equations, and arise in the derivation of model equations describing processes involving rates of change, as in, for example ... 

Equalizing the first derivatives of function (5.26) to zero, we obtain ... 

Derivations of Functions of pH and Ionic Strength that Yield Transformed Thermodynamic Properties at 298.15 K... 

Derivations of Functions of Temperature, pH, and Ionic Strength for Standard Transformed Properties of Reactants... 

Bennovitsky, E.L. (1988) Derivation of functional dependences for roughness coefficient of partially vegetated beds, Water resources (Wodnije resourci), N 1, 68-74. (in Russian)... 

For equally spaced points, the first derivative of function f(x) is approximated (to error order of Ajc ) as follows ... 

A typical example of a static catastrophe in the system (1.6) is the change in number of solutions of the system of equations (1.11) on varying c (a catastrophe of this type is called bifurcation). The condition of occurrence of a catastrophe of this type is expressed in terms of derivatives of functions U dfi/fyj. 

This feature contrasts with the properties of ordinary dijferential equations (ODEs), which contain only derivatives of functions of a single variable. Usually the form of the solution to an ODE is dictated by the ODE itself. Thus, the linear, first-order ODE describing unimolecular decay ... 

Many of the reactions which lead to derivatives of functionalized phosphonic or phosphinic acid derivatives, and which were described in Chapters 3 and 4 have their parallels in the behaviour of the corresponding higher chalcogen-containing species towards the same substrates. Some of the modified reactions are not well exemplified, although other reactions have been widely used for many years general sources should be consulted for early examples. The following short selection of examples should suffice to indicate how the various reactions have been modified, or could well be adapted in the future. 

We denote —dF/dm2 (the negative derivative of function (2.79)2 of reacting fluid mixture with memory) as the chemical affinity (cf. also (2.94). Because of the simple reaction (2.73) we use mass units here for complicated stoichiometry we prefer molar units, see (4.176)). ... 

Note also that the following relationships are vaUd in the starred frame for the time derivative of function tp(f) (see (3.26), (3.28))... 

Sufficient condition of the minimum (4.309) may be calculated from (a). It gives some limits on equilibrium values of derivatives of functions Jp = Jp T, Py), AP = AP(T, Py). We omit them here for simphcity moreover practically the same limitation is given by (e) of Rem. 22 obtained analogously from (b) and (c) there. 

The function is concave meaning that the point of maximal productivity is characterized by the derivative of function r =J[Cx) being equal to zero. 

Some methods adopt the Jacobian matrix J of the system f since its elements caimot be provided analytically in certain circumstances, it is necessary to approximate them numerically. Conceptually, there are no further difficulties in doing so, since they are the first derivatives of functions with respect to the different variables. However, one numerical problem does raise its head large round-off errors are obtained in one small step, whereas the first derivatives are spoiled by higher order derivatives if a large step is adopted. 

Main objective in this paper is the revision and application of parametric methods in imcertainty propagation. However, before applying these methods we must ensure that output vector really follows a normal multivariate distribution. If no information is available about the joint density function in the output, the widespread procedure to ensure this assumption is by means of a multinormal contrast of goodness of fit. Fortunately, following theorem ensures the as3nnptotic j oint normal distribution o f the output vector, when this normality is fulfilled in the input vector, under some weakly conditions about the differentiability of functions in the simulator, providing also the mean vector and covariance matrix of the output vector as functions of their equivalents parameters in the input and the partial derivates of functions in the simulator. 

In practical applications, the initial conditions are frequently not available, and it may not even be clear what their physical meaning is [48]. Therefore, to avoid this conflict, Caputo [45] has proposed that one should incorporate the integer order (classical) derivative of function x, as they are commonly used in initial value problems with integer-order equations. In that way, one can use the derivatives of the Caputo type such as ... 

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