Definition of the Derivative

There is a formal definition of a smoothly varying function , but for our purposes, what we mean is that the curve has no breaks or kinks. 

We can draw a unique tangent line (a straight line whose slope matches the curve s slope) at each point on the curve. Recall that the slope of a line is defined as the amount y changes if x is changed by one for example, the line y = 3x + 6 has a slope of three. 

Much of the first semester of calculus is devoted to understanding what is meant by a smoothly varying function, and finding the derivatives of various functions. For 

FIGURE 2.1 Graph of an arbitrary function f (x). The dashed lines show tangent curves at several points. The slope of the tangent line (called the derivative) can be found by drawing a line between two very close points (here x0 and x0 + Ax). 

The last two terms are dropped in going from Equation 2.2 to Equation 2.3 because they vanish as Ax approaches zero. We could also just say d(x )/dx = 3x2, leaving out the part which implied that the derivative is actually evaluated at x = xq. 

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The Euler method, while extremely inaccurate, is also extremely simple. This method is based on the definition of the derivative... 

Figure 54-1, however, still shows a number of characteristics that reveal the behavior of derivatives. First of all, we note that the first derivative crosses the X-axis at the wavelength where the absorbance peak has a maximum, and has maximum values (both positive and negative) at the point of maximum slope of the absorbance bands. These characteristics, of course, reflect the definition of the derivative as a measure of the slope of the underlying curve. For Gaussian bands, the maxima of the first derivatives also correspond to the standard deviation of the underlying spectral curve. 

In order to show that the expected-value and derivative operations commute, we begin with the definition of the derivative in terms of a limit 16... 

A finite difference representation for a derivative can be introduced by recalling the definition of the derivative of a function 4> x)... 

The finite difference of derivatives involves the approximation of a differential equation or a boundary condition by algebraic equations. Consider the function T(x) shown in Fig. 5.1. The definition of the derivative of T(x) at Xj is given by... 

What we have here, with the Euler method, is the definition of the derivative as pertaining to time t (or nfit) and thus f(y(t)) or /( . ) on the right-hand side. For our specific example (4.3), this becomes approximately... 

A similar result for cp(s, Hd) but with a different relation for parameter X was obtained in application 4.3.1. The value of the derivative cp (0, Hd) is then obtained using the definition of the derivative in a point ... 

Fig. Z5. A contour map for the NaQ molecule overlaid with trajectories of Vp. With the exception of the four trajectories associated with the (3, — 1) critical point (denoted by a dotX the tiaj ories originate at infinity and terminate at one of the two nuclei. Two trajectories originate at infinity and terminate at the (3, — 1) critical point, while two others originate at this point and terminate, one each, at the nuclei. The property of zero flux in the gradient vectors of p is illustrated for the interatomic surface whose intersection with this plane is given by the two trajectories which terminate at the critical point. An arbitrarily drawn surface is shown not to have this property of zero flux. The definition of the derivative dr/dl as the limit of Ar/A/ is also shown.

This is where the spreadsheet comes in, because we can use it to compute the numerical value of the derivative dF/ dx even when you, my reader, may be uncomfortable (or even unfamiliar) with calculus. Here is the definition of the derivative ... 

This could be related to a commutation relation among the integral operators. Typical relations among the infinitesimal operators can be derived from this approach. He had come close to a derivation of the canonical commutation relation from the definition of the derivative of an operatorvalued function of a real variable. Before this canonical commutation, Bom considered the assumption of a complex domain of numbers ... 

Now that we have defined the derivative, we can see how derivatives of particular functions are found by using the definition of the derivative. 

If the derivative is positive, the value of the function increases with the value of x if the derivative is negative, the value of the function decreases as x increases. If the derivative is zero, the curve of the function has a horizontal tangent the function has a maximum or a minimum value. The fundamental definition of the derivative... 

Using the definition of the derivative for the first two terms, we obtain... 

The definition of the derivative follows from the differential. We multiply and divide df(xo h) by h in the last equation to obtain... 

Let US now consider the first derivative the mathematical definition of the derivative is the ratio of the change in concentration to the change of... 

Using the definition of the derivative, the first two terms in the LHS of Eq. 8.19 become the first derivative of / that is,... 

GRAPH 9.38 Insertion into the Formal Graph of a spatial oscillator of halfway intermediate variables ( semienergies ) allows the definition of the derivation with respect to the phase angle cp. 

Inheritance does not in itself cause any particular difficulty for program analysis, since a class derived by inheritance could be expanded by substituting the member declarations of the inherited class(es) into the definition of the derived class. However, the combination of inheritance, polymorphism and dynamic binding is not directly amenable to traditional static analysis, which typically requires that the target of each procedure call is statically known. 

The agreement vlth experimental values is good except for, ubere the differeiwb lies in the definition of The derivation of dr from the measurement of f is base m the older definition of vhioh is the nuefber of neutrons vhioh slow down the Vr fission threshold for each neutron produced by thermal fission (See Reference 12). An average for the RPR startup element is 0.067. 

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