My calculus is rusty and I don\'t understand how line #1 in the picture above si

Question

My calculus is rusty and I don't understand how line #1 in the picture above simplifies down to line #2. Please explain in the highest number of simplified steps as possible (without doing anything ridiculous like using the definition of the derivative (xh/h etc...)).

Example Problem 6.3 show that ( ,-- Solution Becauseus a state tunction, (W( ).)s-G( )s)v Substituting dU= TdS-PdV in the previous expression #1 VBS 8T #2 The Maxwell relations have been derived using only the property that U, H, A, and G are state functions. These four relations are extremely useful in transforming seemingly obscure partial derivatives in other partial derivatives that can be directly measured. For example, these relations will be used to express U, H, and heat capacities solely in terms of measurable quantities such as K,s and the state variables P, V, and T in Supplemental Section 6.15

Explanation / Answer

from 1st law of thermodynamics, dU= qQ+dW

but dQ= TdS, dW =-PdV (1)

hence dU= TdS- PdV

differentiating the equation with respect to S at constant volume

(dU/dS)V= T

taking the second derviartive with resepct to volume at constant entropy

d/dV( dU/dS) = (dT/dV)S   (2)

Eq. is dU= TdS- PdV

differentiating the equation with respect to V at constant S,

(dU/dV)S= -P

differenting the equation with respect to S at constant volume

d/dS (dU/dV) = (-dP/dS)V   (3)

LHS of Eq.2= LHS of Eq.3 hence

(dT/dV)S= (-dP/dS)V

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