What Mathematical Identity is Used in This Derivation?

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The discussion focuses on the application of the product rule in differential calculus to derive an equation from the left-hand side (LHS) to the right-hand side (RHS). The key mathematical identity mentioned is the rearranged product rule: (\frac{dA}{dt})\cdotB = \frac{d}{dt}(A\cdotB) - A\cdot\frac{dB}{dt}. Participants confirm that applying this identity to the first term on the RHS simplifies the equation back to the LHS, validating the derivation process.

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Consider the equation on the attached picture. What mathematical identity for differential calculus (which I assume is used) is used to go from the LHS to the RHS? I tried invoking the product rule but couldn't get anywhere.
 

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You're on the right track. Apply the product rule to the first term on the RHS and show that the entire RHS reduces to the LHS.
 
Or, if you really want to go from the LHS to the RHS then use ([itex]\frac{dA}{dt}[/itex][itex])\cdot[/itex]B = [itex]\frac{d}{dt}[/itex](A[itex]\cdot[/itex]B) - A[itex]\cdot[/itex][itex]\frac{dB}{dt}[/itex] which is just the product rule rearranged.
 

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