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Proving the product rule

Proving the product rule for derivatives.
The product rule tells us how to find the derivative of the product of two functions:
ddx[f(x)g(x)]=ddx[f(x)]g(x)+f(x)ddx[g(x)]=f(x)g(x)+f(x)g(x)\begin{aligned} \dfrac{d}{dx}[f(x)\cdot g(x)]&=\dfrac{d}{dx}[f(x)]\cdot g(x)+f(x)\cdot\dfrac{d}{dx}[g(x)] \\\\ &=f'(x)g(x)+f(x)g'(x) \end{aligned}
The AP Calculus course doesn't require knowing the proof of this rule, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn.

Without further ado, we present to you the proof!

Khan Academy video wrapper
Product rule proofSee video transcript

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