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I have a very basic knowledge of calculus of one variable .

In the chapter on heat and thermodynamics , ideal gas law PV =nRT is given .

Then the book says, differentiating you get

PdV +VdP = nRdT .

The book doesn't explain the differentiation step .

I think , there are two ways to differentiate the gas law PV =nRT

1) Applying product rule of differentiation when a single variable is involved :

Assuming all the three P, V, T are functions of a common variable x , I can differentiate both sides of PV = nRT by x .

d[P(x)V(x)]/dx = d[nRT]dx

Applying product rule on left side I get ,

VdP/dx+PdV/dx = nRdT/dx

Eliminating dx from the denominator from both sides I get ,

VdP+PdV = nRdT

2) Taking total derivative of both sides ,

d(PV) = d(nRT)

[∂(PV)/∂P]dP + [∂(PV)/∂V]dV = [∂(nRT)/∂T]dT

This also gives PdV +VdP = nRdT

Both approaches give same result(equation) .

Is it the product rule that is applied or is it the total derivative (involving partial differentiation ) being applied here ?

Thank you

In the chapter on heat and thermodynamics , ideal gas law PV =nRT is given .

Then the book says, differentiating you get

PdV +VdP = nRdT .

The book doesn't explain the differentiation step .

I think , there are two ways to differentiate the gas law PV =nRT

1) Applying product rule of differentiation when a single variable is involved :

Assuming all the three P, V, T are functions of a common variable x , I can differentiate both sides of PV = nRT by x .

d[P(x)V(x)]/dx = d[nRT]dx

Applying product rule on left side I get ,

VdP/dx+PdV/dx = nRdT/dx

Eliminating dx from the denominator from both sides I get ,

VdP+PdV = nRdT

2) Taking total derivative of both sides ,

d(PV) = d(nRT)

[∂(PV)/∂P]dP + [∂(PV)/∂V]dV = [∂(nRT)/∂T]dT

This also gives PdV +VdP = nRdT

Both approaches give same result(equation) .

Is it the product rule that is applied or is it the total derivative (involving partial differentiation ) being applied here ?

Thank you

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