Unclear differential equation from a thermodynamics textbook

[NODARman]

Homework Statement

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Relevant Equations

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In the thermodynamics textbook there is written: 𝛿𝐴 = 𝑇𝑑𝑆 − 𝑑𝑈 = 𝑑(𝑇𝑆) − 𝑆𝑑𝑇 − 𝑑𝑈 = −𝑑(𝑈 − 𝑇𝑆) − 𝑆𝑑𝑇 = −𝑑𝐹 − 𝑆𝑑𝑇
How did we get the bolded area from TdS? Is that property of derivative, integral, or something else :/

 

[vela]

Product rule

 

[vela]

Sorry for being so terse. I was posting from my phone.

Apply the product rule to $TS$. $d(TS) = ?$.