Does the definition field change after derivative derivation

Does the definition field change after derivative derivation

The definition domain of the function is different from that of the derivative function after derivation. Take an example to know:
Y = under the third root sign (x - 1)
Here x can be 1;
dy/dx = (1/3)(x - 1)^(-2/3)
Definition domain of derivative dy / DX, X ≠ 1

How to determine the extreme value of partial derivative? For example, given that a, B and C are positive numbers satisfying a ^ 2 = B ^ 2 + C ^ 2, find the minimum value of function f (a, B, c) = [a ^ 3 + B ^ 3 + C ^ 3] / [a ^ 2 * (B + C) + B ^ 2 * (a + C) + C ^ 2 * (a + b)], Explain how to use a ^ 2 = B ^ 2 + C ^ 2? Why?

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Using partial derivative to find the extreme value of function z = 1-x2-y2 Such as title

∂z/∂x=-2x
∂z/∂y=-2y
Let ∂ Z / ∂ x = 0 ∂ Z / ∂ y = 0 to obtain: x = 0 y = 0
∂2z/∂x2=-2 ∂2z/∂x∂y=0 ∂2z/∂y2=-2
At (0,0) a = - 20
(0,0) is the maximum point, and the maximum is Z (0,0) = 1

How to deduce the basic equations of thermodynamics Basic equations of thermodynamics dU=TdS－pdV dH=TdS+Vdp dA=－SdT－pdV

In the reversible cycle, the entropy total differential DS = DQ / T, which is determined by the first law of thermodynamics DQ = DU + DW; If only the work done by the change of system volume is considered, then DW = PDV, which is brought into the first law, TDS = DU + PDV, that is, Du = TDS PDV; And because H = u + PV (Legendre transformation), DH = DU + PDV + VDP, added with the above formula, there is

On thermodynamic equations (P + an ^ 2 / V ^ 2) (V-Nb) = NRT my question is that in the former equation, P is the pressure of real gas, and an ^ 2 / V ^ 2 is the internal pressure, so the sum is the ideal pressure, but in the latter equation, V-Nb is the volume of real gas, because the ideal gas volume V minus the volume of each molecule itself, Nb is the real volume, which feels inconsistent, The pressure of the ideal gas x the real gas volume = RT, I think it should be p-an ^ 2 / V ^ 2

P and V in van der Waals equation are the actual values of pressure and volume. V-Nb is the space where gas molecules move freely, that is, it is equivalent to the volume of ideal gas without the volume of molecule itself, so there is no contradiction. The real gas volume includes the volume of molecule itself, and the ideal gas volume does not include the volume of molecule itself (in fact, the volume of molecule itself is relatively small and can be ignored)

Exercises of the second law of thermodynamics A system absorbs 473 J of heat reversibly from a hot reservoir at 529 K and gives off 182.1 J of heat to a cold reservoir at 213 K.During this process,124.6 J of work is done by the system 473j heat is absorbed from the heat source reversibly at 529k, and 182.1j heat is released to the cold object at 213k. 124.6j work is done in this process Find the process 1. The change of internal energy J 2. The change of entropy J / K 3. The change of cosmic entropy

The change of internal energy is:
473-182.1-124.6
Here, the change of system entropy: the heat release of 529k heat source Q1 = 473, the entropy change is - 473 / 529213k, the entropy change of heat source is positive, 182.1/213, and the sum of the two is the total entropy change
Then the entropy change of the universe is 0. Because it is a reversible process