Second differential coefficient

This gives (L//G )v/(pg/Pl) = 16, when dG2F/dpG = 0- As the second differential coefficient is negative at this point, G2F is a maximum. 

Product Rule y = uv differential coefficient =first factor (u) multiplied by differential of second factor (dv/dx)+ second factor (v) multiplied by the differential of the first factor (du/dx) dy/dx = u.dv/dx + v. du/dx ... 

The capacity C of the double layer per square centimetre, or the rate of change of the charge with the applied potential E, can be found from the electrocapillary curve, for it is equal to the second differential coefficient of the surface tension with respect to the potential. This follows at once from (13), since... 

A further thermodynamic expression for l is possible.4 Since the electrocapillary equation (Eq. 8) is a total differential equation, the second cross-partial-differential coefficients of y are equal ... 

The effective diffusion coefficients were calculated from the experimentally observed data (time, amount of cation exchanged, temperature), using Paterson s solution of Fick s second law, or published approximate solutions (8, 16). Taking into consideration particle shape and particle size distribution, the differential coefficients of internal diffusion in ion exchange can be ascertained by a method previously described (9). 

The second differential coefficients of jpa and fi, are thus both positive, and hence both curves are convex to the i/ axis. 

U, H, and S as Functions of T and P or T and V At constant composition, molar thermodynamic properties can be considered functions of T and P (postulate 5). Alternatively, because V is related to T and P through an equation of state, V can serve rather than P as the second independent variable. The useful equations for the total differentials of U, H, and S that result are given in Table 4-1 by Eqs. (4-22) through (4-25). The obvious next step is substitution for the partial differential coefficients in favor of measurable quantities. This purpose is served by definition of two heat capacities, one at constant pressure and the other at constant volume ... 

St is made smaller and smaller without limit but we constantly find that dx/dt is used when Sx/St is intended. For convenience, D is sometimes used as a symbol for the operation in place of djdx. The notation we are using is due to Leibnitz 2 Newton, the discoverer of this calculus, superscribed a small dot over the dependent variable for the first differential coefficient, two dots for the second, thus x, x. .. 

In words, (17) may be expressed the differential coefficient of a function with respect to a given variable is equal to the product of the differential coefficient of the function with respect to a second function and the differential coefficient of the second function with respect to the given variable. We can get a physical meaning of this formula by taking x as time. In that case, the rate of change of a function of a variable is equal to the product of the rate of change of that function with respect to the variable, and the rate of change of the variable. 

The differential coefficient derived from any function of a variable may be either another function of the variable, or a constant. The new function may be differentiated again in order to obtain the second differential coefficient. We can obtain the third and higher derivatives in the same way. Thus, if y = a 3,... 

Just as the first derivative of x with respect to t measures a velocity, the second differential coefficient of x with respect to t measures an acceleration (page 17). For instance, if a material1... 

Examples.—(1) If a body falls from a vertical height according to the law s = gt2, where g represents the acceleration due to the earth s gravity, show that g is equal to the second differential coefficient of s with respect to t. 

Fortunately, in applying the calculus to practical work, only the first and second derivatives are often wanted, the third and fourth but seldom. The calculation of the higher differential coefficients may be a laborious process. Leibnitz s theorem, named after its discoverer, helps to shorten the operation. It also furnishes us with the general or nth derivative of the function which is useful in discussions upon the theory of the subject. We shall here regard it as an exercise upon successive differentiation. The direct object of Leibnitz s theorem is to find the nth differential coefficient of the product of two functions of x in terms of the differential coefficients of each function. 

Hence a curve is concave or convex upwards, according as the second differential coefficient is positive or negative. 

I have assumed that the curve is on the positive side of the -axis when the curve lies on the negative side, assume the a -axis to be displaced parallel with itself until the above condition is attained. A more general rule, which evades the above limitation, is proved in the regular text-books. The proof is of little importance for our purpose. The rule is to the effect that a curve is concave or convex upwards according as the product of the ordinate of the curve and the second differential coefficient, i.e., according as yd y/dx2 is positive or negative . 

Hence the rule In order to find a point of inflexion we must equate the second differential coefficient of the function to zero find the value of x which satisfies these conditions and test if the t second differential coefficient does really change sign by substituting in the second differential coefficient a value of a a little greater and one a little less than the critical value. If there is no change of sign we are not dealing with a point of inflexion... 

The second differential coefficient of the curve (y - g2)2 = g5 has two different values of the same sign. The cusp is then said to be a cusp of the second species, or a rhamphoid cusp. The lower curve also has a maximum when x — The general form of the curve is shown in Fig. 83. 

Let us now see what we can learn from the second differential coefficient... 

The numerator of the first fraction on the right is the differential coefficient of the denominator and hence, its integral is log (a+by+ya) the integral of the second term of the right member is got by the addition and subtraction of ib in the denominator. Hence,... 

Just as it is sometimes necessary, or convenient, to employ the second, third or the higher differential coefficients dPy/da ..., so it is often just as necessary to apply successive integration to reverse these processes of differentiation. Suppose that it is required to reduce, dPyjdx = 2, to its original primitive form. We can write for the first integration... 

But h2, being the square of a number, must be positive. The sign of the second differential coefficient will, in consequence, be the same as that of Sm. But u = f(x) is a maximum or a minimum according as hi is negative or positive. This means that y will be... 

If, however, the second differential coefficient vanishes, the reasoning used in connection with the first differential must be applied to the third differential coefficient. If the tjiird derivative vanishes, a similar relation holds between the second and fourth differential coefficients. See Table I., page 168. Hence the rules —... 

The student may now show that by differentiating Stirling s formula twice, and putting x = 0 in the result, we obtain the second differential coefficient... 

A differential equation is ordinary or partial, according as there is one or more than one independent variables present. Ordinary differential equations will be treated first. Equations like (2) and (3) above are said to be of the first order, because the highest derivative present is of the first order. For a similar reason (4) and (6) are of the second order, (5) of the third order. The order of a differential equation, therefore, is fixed by that of the highest differential coefficient it contains. The degree of a differential equation is the highest power of the highest order of of differential coefficient it contains. This equation is of the second order and first degree ... 

Most of your troubles in connection with this branch of the calculus of variations will arise from this equation. It is often very refractory sometimes it proves too much for us. The equation then remains unsolved. The nature of the problem will often show directly, without any further trouble, whether it be a maximum or a minimum value of the function we are dealing with if not, the sign of the second differential coefficients must be examined. The second derivative is positive, if the function is a minimum and negative, if the function is a maximum. But you will have to look up some text-book for particulars, say B. Williamson s Integral Calculus, London, 463, 1896. 

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