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The product rule (2 proofs) and the quotient rule (1 proof).
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The Derivative of a Product of Functions: The Product Rule
Proof: The classic proof: Adding and subtracting the same value is the same as adding zero
The Derivative of a Product of Functions: The Product Rule
Proof: The classic proof: Adding and subtracting the same value is the same as adding zero
A more non-traditional proof: